WEBVTT

1
00:00:00.000 --> 00:00:06.240
 Well hello everyone and welcome to our second class on Laplace transforms. Today we're going to

2
00:00:06.240 --> 00:00:12.640
 introduce the inverse Laplace transform and we're going to see how we can use the first

3
00:00:12.640 --> 00:00:18.320
 shifting theorem in reverse, in other words use the first shifting theorem to calculate

4
00:00:18.320 --> 00:00:24.720
 inverse Laplace transforms and we'll also look at inverse Laplace transforms requiring algebraic

5
00:00:24.720 --> 00:00:31.680
 techniques such as partial fractions and complete in the square. Now if you remember in our last

6
00:00:31.680 --> 00:00:39.040
 class we saw this diagram here and we said that Laplace transforms are often used to solve problems

7
00:00:39.040 --> 00:00:45.920
 like differential equations where we're unable to go directly from the calculus problem as a

8
00:00:45.920 --> 00:00:52.960
 differential equation to its solution in the time domain, either because the problem is very difficult

9
00:00:52.960 --> 00:01:00.320
 or even impossible using the methods that we know to solve differential equations. So what we do is

10
00:01:00.320 --> 00:01:06.800
 we take our time domain problem typically at differential equation as I've said take its Laplace

11
00:01:06.800 --> 00:01:15.120
 transform so move from the time domain the t domain over to the s domain and in so doing we convert

12
00:01:15.120 --> 00:01:21.840
 the calculus problem to an algebra problem in the s domain also called the Laplace domain.

13
00:01:22.800 --> 00:01:29.280
 The algebraic problem can be solved without too much bother to get our solution in the

14
00:01:29.280 --> 00:01:38.240
 s domain and then we apply the inverse Laplace transform to go back to the time domain where

15
00:01:38.240 --> 00:01:45.600
 the problem was originally set. So as I said in a previous lecture we looked at this step here

16
00:01:45.600 --> 00:01:51.440
 going from the time domain to the s domain by calculating the Laplace transform. Today we're

17
00:01:51.440 --> 00:01:58.640
 going to go the other way we're going to take our s domain function which could be the solution to

18
00:01:58.640 --> 00:02:06.080
 our algebra problem take our s domain function and take its inverse Laplace transform to go back

19
00:02:06.080 --> 00:02:15.680
 to the time domain. So that will then have completed the time domain to the s domain calculation

20
00:02:16.480 --> 00:02:22.560
 and when we finish today we'll be able to do the s domain back to the t domain and in the next

21
00:02:22.560 --> 00:02:28.800
 lecture we'll look at this bit here solving the algebra problem we'll take a differential equation

22
00:02:28.800 --> 00:02:36.000
 takes Laplace transforms solve the algebra problem and then invert it so but for now we're going to

23
00:02:36.000 --> 00:02:43.280
 focus on calculating inverse Laplace transforms. So the inverse Laplace transform well for a function

24
00:02:43.360 --> 00:02:52.240
 f of s in the s domain its inverse Laplace transform is a function f of t in the t domain

25
00:02:52.240 --> 00:02:59.760
 such that the Laplace transform of f of t is equal to f of s. So in other words last time we saw how

26
00:02:59.760 --> 00:03:05.120
 to calculate this we saw how to calculate the Laplace transform of a time domain function to

27
00:03:05.120 --> 00:03:14.560
 obtain f of s today we are going to take f of s and retrieve the f of t for that f of s. So that's

28
00:03:14.560 --> 00:03:21.600
 what I say here given a Laplace transform of f of s to determine the corresponding f of t we could

29
00:03:21.600 --> 00:03:29.120
 use the inverse transformation and the inverse transformation is given by this integral here

30
00:03:30.080 --> 00:03:34.160
 now in the last lecture we saw that when we're going the other way from the time

31
00:03:34.160 --> 00:03:40.640
 domain to the s domain there was an integral to calculate the Laplace transform it was an

32
00:03:40.640 --> 00:03:46.480
 improper integral with infinity as one of the limits and we saw that it was quite an involved

33
00:03:46.480 --> 00:03:52.240
 calculation that would just become harder and harder as our time domain functions got more

34
00:03:52.240 --> 00:04:01.040
 complicated and that's why we use tables this is even worse because to evaluate this integral we would

35
00:04:01.040 --> 00:04:10.240
 no have to know a topic called complex analysis and it's beyond the scope of this course so what

36
00:04:10.240 --> 00:04:15.520
 we're going to do is we are actually going to take the tables that we're using the last time

37
00:04:15.520 --> 00:04:24.400
 and we're going to use them in reverse so in other words or in lecture one we took a time

38
00:04:24.400 --> 00:04:31.680
 domain function as in this column here and we calculated the Laplace transform using the tables

39
00:04:32.720 --> 00:04:38.800
 today we're going to take our estimate function and we're going to go back the other way so we'll

40
00:04:38.800 --> 00:04:45.200
 have a function in the s domain which is in this column here on the table and we want to retrieve

41
00:04:45.600 --> 00:04:54.320
 the time domain function in the column on the left here now as I've mentioned before this will

42
00:04:55.120 --> 00:05:02.880
 undoubtedly require us to do some manipulation on the time that we are given we need to manipulate it

43
00:05:02.880 --> 00:05:10.560
 to obtain something that's in the form given in the tables so we'll need to know algebraic

44
00:05:10.560 --> 00:05:19.440
 techniques such as completing the square and partial fractions and I'll certainly go into some

45
00:05:19.440 --> 00:05:28.400
 examples of these today so first of all before we launch into some examples let's just have a look at

46
00:05:29.280 --> 00:05:35.440
 the linearity property of the inverse Laplace transform very similar to the linearity property

47
00:05:35.440 --> 00:05:42.640
 of the Laplace transform we saw the other day basically what it's saying is if you've got let's say

48
00:05:42.640 --> 00:05:47.840
 two three it doesn't matter how many functions I've just taken two here you've got two functions in

49
00:05:47.840 --> 00:05:53.600
 the s domain f of s and g of s and they're both multiplied by some constants and you want the Laplace

50
00:05:53.600 --> 00:06:00.800
 of that whole thing well it's perfectly okay to take the constants outside the Laplace the inverse

51
00:06:01.280 --> 00:06:07.920
 operators take the Laplace of inverse Laplace of each individual function and add them together

52
00:06:08.720 --> 00:06:14.880
 so this is very similar to what we did last time and very similar to what we did when we're doing

53
00:06:14.880 --> 00:06:21.200
 differentiation we were differentiating let's say t squared plus sine t we differentiate t squared

54
00:06:21.200 --> 00:06:26.320
 we differentiate sine t and would add the derivatives well the same thing applies here

55
00:06:27.280 --> 00:06:33.760
 we just take in inverse Laplace of this function the inverse Laplace of that function and add them

56
00:06:33.760 --> 00:06:39.280
 together that's all I'm saying here here I've taken the functions the s domain functions

57
00:06:41.040 --> 00:06:46.800
 without being multiplied by a constant so I've got the inverse Laplace of the sum of f of s plus g

58
00:06:46.800 --> 00:06:52.480
 of s that's equal to the inverse Laplace of f of s plus the inverse Laplace of g of s and I've

59
00:06:52.480 --> 00:06:59.440
 got a constant multiplying one of them or both of them we can take the constant outside the operator

60
00:06:59.440 --> 00:07:05.600
 just take the inverse Laplace of f of s and multiply the count by the constant c and that's all

61
00:07:05.600 --> 00:07:15.680
 summarized in one sentence here by this term up at the top of the slide okay so let's have a look

62
00:07:15.760 --> 00:07:27.760
 at calculating some inverse Laplace transforms so the first one is we're given an s domain function

63
00:07:27.760 --> 00:07:35.920
 f of s is equal to six over s to the power four and we want to find what was the time domain function

64
00:07:35.920 --> 00:07:45.120
 whose Laplace transform is equal to six over s to the four okay so like I said we use the tables for

65
00:07:45.120 --> 00:07:53.680
 this so what we do is we look down in this right hand column I focus on the denominator to start

66
00:07:53.680 --> 00:08:00.720
 with so I'm going to focus on the denominator here s to the power four is there anything that looks

67
00:08:00.720 --> 00:08:07.360
 like that in the table well we've got s to the power one here s to the power two s to the power three

68
00:08:08.000 --> 00:08:14.720
 but we've run out there we don't have s to the power four but what we do have is we've got a general

69
00:08:14.800 --> 00:08:26.480
 term here with s to the power n plus one and in our example the n plus one must equal four because

70
00:08:26.480 --> 00:08:32.960
 it's got s to the four the denominator here with s to the n plus one here so n plus one must equal

71
00:08:32.960 --> 00:08:41.760
 four in other words n must equal three so I can now check also what the top line says the top line

72
00:08:41.760 --> 00:08:48.080
 must have n factorial or three factorial three factorial is six and that's exactly what we've

73
00:08:48.080 --> 00:08:55.280
 got in the top line so I've been able to go to the second column or the third column in the table

74
00:08:55.280 --> 00:09:02.400
 the far right column in the table I've compared the denominator and from that I've found that n is equal

75
00:09:02.400 --> 00:09:12.720
 to three I've then had to compare the numerator to see if n factorial matches the value in my question

76
00:09:13.360 --> 00:09:20.560
 and n factorial with n equal to three n factorial is six and that exactly matches what I've got here

77
00:09:20.560 --> 00:09:27.920
 so I can directly use the table here so that tells me that if n is equal to three the inverse

78
00:09:28.000 --> 00:09:36.080
 Laplace of six over s to the power four is equal to the inverse Laplace of three factorial over s

79
00:09:36.080 --> 00:09:43.200
 to the power four and directly with n equal to three I find that that's equal to t cubed and if you

80
00:09:43.200 --> 00:09:50.720
 wanted to check this you could just go and calculate the Laplace transform of t cubed and you should

81
00:09:50.720 --> 00:09:58.000
 get six over s to the power four then you can see that t cubed would be three factorial which is six

82
00:09:58.000 --> 00:10:02.960
 over s to the three plus one which is four you'd have six or s to the four and that's exactly

83
00:10:02.960 --> 00:10:12.000
 what you start with so we're happy that your answer is correct so moving on let's have a look at number two

84
00:10:12.880 --> 00:10:24.640
 in this question we're asked to find the inverse Laplace transform over 15 where s to the power six

85
00:10:26.160 --> 00:10:33.680
 so we want to find which walk was the function f of t when we took its Laplace transform we obtained

86
00:10:33.680 --> 00:10:41.920
 this quantity here this function here so again I would look at my tables focusing on the denominator

87
00:10:42.160 --> 00:10:47.840
 to start with so I've got s to the power six well we know we don't have that in our tables we've

88
00:10:47.840 --> 00:10:56.560
 run out at s cubed but we know that we can use number four so number four is s to the n plus one

89
00:10:56.560 --> 00:11:04.560
 so we match that with the s to the power six so n plus one must equal six so in other words

90
00:11:04.560 --> 00:11:11.440
 n must equal five because that would give you s to five plus one which says to the six so we've got

91
00:11:11.440 --> 00:11:19.680
 s to the power six and we know it's going to be number four with n equal to five now this table

92
00:11:19.680 --> 00:11:30.240
 tells me that if I'm going to use number four that my top line must be five factorial it's

93
00:11:30.240 --> 00:11:36.160
 got to be because I've said from the denominator I've said that to give me s to the power six I

94
00:11:36.160 --> 00:11:43.600
 must have n equal to five and if I am using number four then I must have five factorial in the top

95
00:11:43.600 --> 00:11:52.160
 line so five factorial is 120 and that's not what I've got I've got 15 so I need to do a little bit

96
00:11:52.160 --> 00:12:00.160
 of algebraic manipulation and the easiest way to do this is to write down in the top line exactly

97
00:12:00.240 --> 00:12:07.840
 what the table would have you use in the top line the table wants a five factorial in the top line

98
00:12:08.640 --> 00:12:17.840
 so we write that in and we've got to multiply this by some fractional expression that will

99
00:12:18.720 --> 00:12:24.480
 make sure things aren't changed so that we so we've got the function that we started out with

100
00:12:24.480 --> 00:12:32.160
 and the easiest way to do that is for a fraction to have in the top line we write the thing that we

101
00:12:32.160 --> 00:12:40.160
 want which is 15 and on the bottom line we write what the table has forced us to introduce which is

102
00:12:40.160 --> 00:12:46.400
 five factorial and you can see the five factorial there cancels with a five factorial there leave me

103
00:12:46.400 --> 00:12:56.080
 15 over s to the six and that's exactly what I started out with so I can now go ahead I can do the

104
00:12:56.080 --> 00:13:05.680
 inrational plus of this term here so five factorial or s the power six will just be number four with

105
00:13:05.680 --> 00:13:15.680
 n equal to five so that will be t to the power five 15 over five factorial well five factorial is 120

106
00:13:15.680 --> 00:13:24.720
 so I get this expression here I can easily tidy this up because 15 goes out into 128 times so I've

107
00:13:24.720 --> 00:13:30.800
 got one over eight t to the five and again I could just take the Laplace transform off this

108
00:13:31.520 --> 00:13:42.240
 using number four with n equal to five and I should obtain 15 over s to the six if I do that

109
00:13:42.320 --> 00:13:49.360
 I could check my answer for me so I'm a little bit of manipulation needed there because the

110
00:13:49.360 --> 00:13:56.400
 function we were given isn't exactly as what's given in the table we looked at the denominator

111
00:13:57.280 --> 00:14:03.840
 to try and figure out which one we should use and we we reached number four and we said well

112
00:14:03.840 --> 00:14:11.200
 number four with n equal to five will work but that means I must have five factorial in the top line

113
00:14:11.200 --> 00:14:18.400
 not a problem we wrote that down in the top line and we introduced this factor this important factor

114
00:14:18.400 --> 00:14:24.880
 to make sure that we aren't changing any and in this factor on the top line is the thing that we want

115
00:14:24.880 --> 00:14:32.320
 which is 15 and down below is what the table forces to introduce that's five factorial as you can see

116
00:14:32.320 --> 00:14:40.240
 they cancel to leave me 15 over s to the six which is what I started out with okay moving on let's

117
00:14:40.320 --> 00:14:49.760
 have a look at another example here so we've got f of s is equal to eight over s to the minus three

118
00:14:49.760 --> 00:14:56.800
 and we want to find the inverse Laplace transform of this function and that will give us the function

119
00:14:56.800 --> 00:15:05.600
 f of t which Laplace transform is eight over s to the minus three so f of t is equal to

120
00:15:06.560 --> 00:15:12.560
 inverse Laplace of eight over s minus three so I need to look at my table focus on the denominator

121
00:15:12.560 --> 00:15:19.520
 to start with s minus three well none of these will work but when I hit this line here looking at

122
00:15:19.520 --> 00:15:27.200
 the denominator I've got s plus some number well it doesn't matter that that's a minus because

123
00:15:27.920 --> 00:15:34.560
 you know I could write this as s plus minus three so we can certainly handle that so

124
00:15:35.680 --> 00:15:47.360
 what do we do here so I would say that this one comparing this with my tabulated value I would

125
00:15:47.360 --> 00:15:57.200
 say that this here we would have to have alpha equal to minus three and if alpha is equal to minus

126
00:15:57.280 --> 00:16:08.160
 three then I would get a I would get I've got an eight here eight e to the minus minus three t

127
00:16:08.160 --> 00:16:15.600
 which is eight e to the positive three t and again I could calculate the Laplace transform of this

128
00:16:15.600 --> 00:16:23.520
 to obtain the function f of s that I was given so once again we focus on the denominator

129
00:16:24.480 --> 00:16:30.880
 obtain the candidate from the tables which was number five then we compare the denominator

130
00:16:30.880 --> 00:16:38.560
 we've got s minus three and s plus alpha and that tells me that alpha must be equal to minus three

131
00:16:38.560 --> 00:16:45.280
 so if I put it in here I've got e to the minus minus three t or e to the three t and don't forget

132
00:16:45.280 --> 00:16:52.080
 though there's an eight on the outside here and that gives me my answer so I'll just say one other

133
00:16:52.160 --> 00:17:02.960
 thing here um our table gives us e to the minus alpha t if you had another set of tables they may

134
00:17:02.960 --> 00:17:10.240
 very well have given you e to the plus alpha t and in that case the Laplace transform would be one over

135
00:17:10.800 --> 00:17:17.200
 s minus alpha and we could have just used this one directly like so but of course the tables are

136
00:17:17.200 --> 00:17:25.760
 limited in how much they can contain just as in this year they stopped at s cubed the denominator

137
00:17:26.320 --> 00:17:33.520
 because the tables have to be a sensible size of course so okay so that's another one done so

138
00:17:33.520 --> 00:17:41.760
 let's move on and look at this one here now so we've got three over s times s plus six so once

139
00:17:41.840 --> 00:17:50.560
 again we focus on the denominator we go to the tables well it's none of these s into s plus some

140
00:17:50.560 --> 00:17:58.720
 number well eventually we hit this one here number seven that looks that looks like a possible candidate

141
00:17:59.840 --> 00:18:05.200
 there are others of course after this but you know we rule them out this is the first one

142
00:18:05.200 --> 00:18:11.200
 that we've reached that we think might work so let's have a look at it so what is this one

143
00:18:11.200 --> 00:18:20.080
 required so I want my function f of t will be equal to the inverse Laplace of three over s to the s

144
00:18:20.080 --> 00:18:31.200
 plus six now compare the expression we're given with the expression in the final column with

145
00:18:31.200 --> 00:18:40.240
 f number seven there's a slight difference s into s plus six that's fine the denominator is fine s

146
00:18:40.240 --> 00:18:44.960
 into s plus some number that's exactly what we've got but if you look closely at this

147
00:18:46.080 --> 00:18:53.280
 whatever the number of the top line must match the number here so the alphas must match in our case

148
00:18:53.280 --> 00:18:59.520
 they don't we've got a six down below and a three up top so we've got to do a little bit of algebraic

149
00:18:59.520 --> 00:19:07.440
 manipulation that will allow us to use number seven and we can't really change anything down below

150
00:19:07.440 --> 00:19:13.680
 here we can't change anything down below here because this shift value here is fixed in the

151
00:19:13.680 --> 00:19:20.000
 denominator we can't change that but we can manipulate the top line a little so we use

152
00:19:20.000 --> 00:19:27.280
 exactly the same trick as we did for the previous example we write down here we write down exactly

153
00:19:27.360 --> 00:19:35.360
 what the table would have if i'm going to use number seven so on the denominator i've got s

154
00:19:35.360 --> 00:19:43.840
 into s plus alpha which in our case is s into s plus six now the top line the alpha and the

155
00:19:43.840 --> 00:19:50.880
 top must match the alpha on the bottom so in our case we must have a six in the top line

156
00:19:51.840 --> 00:19:58.960
 and then the two values the alphas in our case which is equal to six will match but in doing that

157
00:19:58.960 --> 00:20:05.360
 we have actually changed what we started out with we started out with a three so we do the same

158
00:20:05.360 --> 00:20:11.920
 trick as in the previous example we multiply by a fraction whose top line is the thing that we want

159
00:20:11.920 --> 00:20:18.800
 which is a three and whose bottom line includes what the table forced us to introduce and that was

160
00:20:18.800 --> 00:20:29.440
 the six so if i do that i can now directly apply the table to this expression here so that is just

161
00:20:29.440 --> 00:20:37.520
 number seven with alpha equal to seven so i get one minus e to the minus seven t for this part here

162
00:20:37.520 --> 00:20:44.320
 this is the inverse part here one minus e to the minus six t sorry one minus e to the minus six t

163
00:20:44.320 --> 00:20:50.560
 because alpha is equal to six and on the outside don't forget we multiplied by this fraction which

164
00:20:50.560 --> 00:20:58.480
 simplifies to a half so that is the answer there that is the function f of t which is a plus transform

165
00:20:58.480 --> 00:21:06.480
 is given by big f of s up here and once again if you wanted to you could calculate the la plus

166
00:21:06.480 --> 00:21:13.840
 transform of this one here from these tables and you would find that you get exactly what's given

167
00:21:13.920 --> 00:21:19.840
 in the question i mean that is fairly straightforward here because if you compare let's just ignore the

168
00:21:19.840 --> 00:21:24.880
 half for now but you look at this part here that's exactly the same as number seven here with alpha

169
00:21:24.880 --> 00:21:33.280
 equal to six so you would get six over s into s plus six multiplied by a half and the six

170
00:21:33.280 --> 00:21:38.480
 the half multiplied again to give you three and that's exactly what you've started out with

171
00:21:38.800 --> 00:21:49.600
 okay so moving on let's have a look at yet another example so what do we have here we've got nine

172
00:21:49.600 --> 00:21:56.880
 over s into s squared plus nine so that's our f of s we want the inverse Laplace of that to see

173
00:21:56.880 --> 00:22:07.280
 which function f of t has this as its Laplace transform so again we focus on the denominator

174
00:22:07.280 --> 00:22:13.440
 and look down the column in our table and see if there's any that looks like this well you know you

175
00:22:13.440 --> 00:22:20.400
 can think of this as s squared plus some number squared you know s squared plus omega squared

176
00:22:20.400 --> 00:22:27.520
 if you like and you know that would give you that omega squared is nine so omega is three

177
00:22:27.520 --> 00:22:32.960
 so let's have a look at the first one is there any of this form in the table well you could work

178
00:22:32.960 --> 00:22:38.400
 all your way all the way down to here that doesn't work it's just s squared plus a number

179
00:22:39.200 --> 00:22:44.960
 that doesn't work it's the same thing but we reached this one here that one could work that

180
00:22:44.960 --> 00:22:53.520
 one could work it's s into s squared plus some number squared called the number omega and that's

181
00:22:53.520 --> 00:23:02.080
 good because we can rewrite this as s into s squared plus three squared meaning that omega is equal to

182
00:23:02.080 --> 00:23:08.800
 three so that's good so we're down let's say we're down about here now we're s into s squared plus

183
00:23:08.800 --> 00:23:15.840
 three squared what is the top line need the top line says that we're going to use number 10 then we

184
00:23:15.840 --> 00:23:22.000
 must have the omega squared we must have this quantity matching the quantity in the denominator

185
00:23:22.000 --> 00:23:28.720
 here and if we look we actually do it's nine up top and it's nine down here they do match they're

186
00:23:28.720 --> 00:23:38.400
 both three squared so omega is equal to three in this case and the inverse Laplace of this function

187
00:23:38.400 --> 00:23:48.160
 the function we started out with is just one minus cos three t that's using omega equal to three and

188
00:23:48.160 --> 00:23:56.320
 again you can check this and I did that here just so that you can see it's the Laplace transform

189
00:23:56.320 --> 00:24:01.760
 of one minus cos three t this is what we're doing in our last lecture you can take because of the

190
00:24:01.760 --> 00:24:06.160
 linearity property you can take the Laplace transform of each individual term and then

191
00:24:06.160 --> 00:24:12.640
 perform the subtraction so if I do that that's one over s that's number one in the table this

192
00:24:13.440 --> 00:24:19.920
 is number nine in the table cos three t gives me that it's s over s squared plus three squared s

193
00:24:19.920 --> 00:24:26.800
 over s squared plus nine I then write this over a common denominator so my common denominator would

194
00:24:26.800 --> 00:24:32.720
 be s times s squared plus nine so if I multiply that by s squared plus nine I must multiply the top

195
00:24:32.720 --> 00:24:39.120
 by s squared plus nine that's where that comes from and here I'm multiplying the denominator by s

196
00:24:40.080 --> 00:24:47.280
 so I must multiply the top by s so I get s squared I simplify all of that and I get back exactly what I started out with

197
00:24:48.080 --> 00:24:57.360
 so one minus cos three t is the function f of t which Laplace transform is nine over s into s squared plus nine

198
00:24:58.640 --> 00:24:59.760
 okay um

199
00:25:02.880 --> 00:25:06.800
 moving on let's have a look what we've got here

200
00:25:09.680 --> 00:25:19.760
 okay 13 over s squared plus three right so again we'll focus on the denominator and see is there

201
00:25:19.760 --> 00:25:26.800
 anything in the tables that looks like that well I would reach number eight I've got s squared plus

202
00:25:26.800 --> 00:25:32.960
 omega squared so it's basically s squared plus a number and that's exactly what I've got so it could

203
00:25:32.960 --> 00:25:41.040
 be number eight I move on ah it could also be number nine s squared plus a number so

204
00:25:42.080 --> 00:25:48.000
 it's going to be one of those two and how do I know which one it is well I now look at the top

205
00:25:48.000 --> 00:25:54.720
 line I look at the numerator and first of all when I look at number nine I see number nine

206
00:25:54.720 --> 00:26:04.080
 includes the Laplace variable s whereas number eight in the top line just includes a constant a number

207
00:26:05.680 --> 00:26:14.800
 the Laplace variable s doesn't appear and I look at my question and I see there's no s up top so that

208
00:26:14.800 --> 00:26:22.240
 tells me with just a number in the top line that tells me that I should use number eight so it's

209
00:26:22.240 --> 00:26:28.400
 going to be number eight but I need to do a little bit of work first of all it needs a little bit of

210
00:26:28.400 --> 00:26:36.400
 manipulation so my function f of t is the inverse Laplace of 13 over s squared plus three that's what

211
00:26:36.400 --> 00:26:48.320
 I've been given and I've identified that it's going to be number eight that I'm going to use here

212
00:26:49.040 --> 00:26:56.240
 and that should actually read that omega squared is equal to three I'll just make that correction

213
00:26:56.240 --> 00:27:02.720
 right now I think I maybe missed out the square on this term here so that should read

214
00:27:03.920 --> 00:27:10.080
 omega squared is equal to three so let's do a little bit of manipulation to get

215
00:27:10.960 --> 00:27:15.760
 this correct so I've actually got it written correctly in green here omega squared is equal to

216
00:27:16.560 --> 00:27:26.960
 so that means that omega will be root three so I can rewrite this term here as s squared plus

217
00:27:26.960 --> 00:27:32.800
 omega squared or in other words s squared plus root three squared and that'll give me my s squared

218
00:27:32.800 --> 00:27:37.840
 plus three because when you square the square root it just gives you three now what about the top

219
00:27:37.840 --> 00:27:49.040
 line I've got to be careful here my top line at 13 that's not equal remember to omega omega we said

220
00:27:49.040 --> 00:27:56.800
 was root three so what I'm what I need to have is I must have a root three in the top line here so

221
00:27:56.800 --> 00:28:03.040
 that's what I'm going to do I'm going to write down exactly what the table would force me to have

222
00:28:03.040 --> 00:28:12.080
 in the top line and now I've got the form omega over s squared plus omega squared exactly this form

223
00:28:12.800 --> 00:28:19.200
 but I've changed things so I must have my correction factor in here again it's this fraction

224
00:28:19.200 --> 00:28:26.640
 whose top line is the thing that I want and that's a 13 and down below here is what the table

225
00:28:26.640 --> 00:28:33.600
 forced me to introduce and that was root three so as you can see if I cancel out the root threes

226
00:28:33.600 --> 00:28:40.960
 I'll get 13 on the top line it's exactly what I started out with and I can now use the tables

227
00:28:40.960 --> 00:28:50.400
 directly on this expression here with number eight and in that case omega is equal to root three so

228
00:28:51.360 --> 00:29:00.720
 the inverse Laplace of this thing is just sine root three t and of course don't forget the factor

229
00:29:00.720 --> 00:29:10.000
 on the outside 13 over root three so that is the function f of t which Laplace transform is given

230
00:29:10.000 --> 00:29:19.840
 by 13 over s squared plus three okay next problem so our next problem will be given our function

231
00:29:19.920 --> 00:29:32.320
 f of s is equal to four over s seven s plus three so we're looking for the function f of t whose Laplace

232
00:29:32.320 --> 00:29:42.480
 transform is given by four over seven s plus three now again we look through the table focusing on

233
00:29:42.480 --> 00:29:49.200
 the denominator and I don't I won't find this I will not find this in the table because as I've said

234
00:29:50.640 --> 00:29:57.840
 before the tables of course have to be a manageable size and we can't include every

235
00:29:57.840 --> 00:30:05.440
 single type of function in the table so we only include some general terms and we need to do some

236
00:30:05.440 --> 00:30:12.480
 manipulation to get this into a tabulated form so searching down the table the only one that looks

237
00:30:12.560 --> 00:30:21.760
 like it would be this one but s plus alpha where the coefficient of s is one and if you look in the

238
00:30:21.760 --> 00:30:28.160
 table the coefficient of s in virtually all cases is equal to one we've got a seven front

239
00:30:28.160 --> 00:30:35.120
 bs but that's not a problem we can manipulate this and get it into the tabulated form so it's

240
00:30:35.120 --> 00:30:41.120
 number five we're going to use here and what we're going to have to do is we're going to have to write

241
00:30:41.200 --> 00:30:49.600
 as a set s plus something it'll all be multiplied by something but that factor doesn't matter let's

242
00:30:49.600 --> 00:30:55.520
 get it written as s plus a number first of all so what I'm going to do here is I'm going to take the

243
00:30:55.520 --> 00:31:02.320
 seven out as a common factor so I'll just get the pin I'll take the seven out as a common factor I've

244
00:31:02.320 --> 00:31:11.360
 got seven s plus three I'll take the seven out as a common factor now to get seven s I need an s there

245
00:31:12.000 --> 00:31:19.040
 plus now plus what well if I wrote three there that wouldn't that wouldn't be right because seven times

246
00:31:19.040 --> 00:31:25.520
 three would be 21 and that's not what I've got so what I'm going to do is if I'm going to multiply

247
00:31:25.600 --> 00:31:33.680
 the three by seven so that I don't change anything I just divide by seven and you can check that it

248
00:31:33.680 --> 00:31:41.040
 can be seven s plus 21 over seven 21 over seven is three and so I haven't changed anything and I've

249
00:31:41.040 --> 00:31:48.960
 got it written here as well so that will be the manipulation that I'll do here I'll have

250
00:31:49.920 --> 00:32:01.040
 four over seven into s plus three over seven so that will be that will be number five and I've

251
00:32:01.040 --> 00:32:08.560
 got the s plus alpha form in the denominator now I can break this up as four over seven times that

252
00:32:08.560 --> 00:32:15.520
 and now I can then have four over seven think it's a constant so I can take it outside the

253
00:32:15.600 --> 00:32:21.280
 inverse Laplace operator I can pull that outside so I just got to get the inverse Laplace of that

254
00:32:21.280 --> 00:32:28.320
 term there that's it I've repeated it down here but that's easy that's easy that's just number five

255
00:32:28.320 --> 00:32:36.640
 with alpha equal to three over seven and the inverse Laplace of that is e to the minus three

256
00:32:36.640 --> 00:32:43.760
 over seven t and don't forget we've got the four sevenths on the outside multiplying that

257
00:32:44.960 --> 00:32:52.240
 so I use number five with alpha equal to three sevenths to give me e to the minus three over seven

258
00:32:52.240 --> 00:32:58.320
 t and then I had my multiplying factor on the outside which was four sevenths and again if you

259
00:32:58.320 --> 00:33:05.600
 wanted to check that that answer is correct you would take the Laplace transform of this function

260
00:33:05.600 --> 00:33:18.640
 here okay so we can move on and look at the next set of calculations of the inverse Laplace transform

261
00:33:21.360 --> 00:33:28.720
 now in our last class we introduced what we called the first shifting theorem or the FST for

262
00:33:29.360 --> 00:33:36.720
 short and we said that the first shifting theorem was used to calculate the Laplace

263
00:33:36.720 --> 00:33:44.880
 transforms of a product involving an exponential function we also said that it was really important

264
00:33:44.880 --> 00:33:52.320
 to remember that if you got the Laplace transform if you're asked for the Laplace transforms of two

265
00:33:52.320 --> 00:34:01.120
 functions of t f t times g of t the Laplace transform of a product it is not equal to the

266
00:34:01.120 --> 00:34:10.720
 Laplace trans the product of the individual Laplace transforms that is incorrect just the same way as

267
00:34:10.720 --> 00:34:18.800
 when we're doing differentiation and you had to differentiate something like t cubed cos t you

268
00:34:18.800 --> 00:34:25.840
 wouldn't just take the Laplace the derivative of t cubed and multiply it by the derivative of cos t

269
00:34:25.840 --> 00:34:35.280
 you had to use the product rule similar things apply for Laplace transforms in that the Laplace

270
00:34:35.280 --> 00:34:42.000
 transform of a product as I've just finished saying is not equal to the product of the individual

271
00:34:42.000 --> 00:34:49.440
 Laplace transforms that's completely wrong and we saw in the last lecture that for a specific

272
00:34:49.440 --> 00:34:56.560
 product where we had an exponential function multiplying some other function at time that we

273
00:34:56.560 --> 00:35:03.440
 had to use the first shifting theorem to obtain the Laplace transform of that specific

274
00:35:04.400 --> 00:35:11.120
 product type and what it amounts to is when you calculate the Laplace transform of this time

275
00:35:11.120 --> 00:35:20.240
 domain function what you get is you get the a shifted function in the estimate basically what I mean

276
00:35:20.240 --> 00:35:29.440
 by that is if your raw function f of t has Laplace transform big f of s then if you multiply your

277
00:35:29.440 --> 00:35:36.320
 function f of t by an exponential function e to the minus alpha t what you need to do to the Laplace

278
00:35:36.320 --> 00:35:46.640
 transform of f of s is shift it by an amount alpha and we saw that this table here allowed us or

279
00:35:46.640 --> 00:35:57.680
 helped us to calculate the Laplace transform of a function that was a product of an exponential

280
00:35:57.680 --> 00:36:05.920
 and some other time domain function so we start down here with the function we're giving we've got

281
00:36:06.160 --> 00:36:11.200
 an e to the minus alpha t multiplying some function f of t it could be t to the power six or it could

282
00:36:11.200 --> 00:36:19.760
 be sine 3t or cos 9t or anything like that so if you remember what we did was we ignored the exponential

283
00:36:19.760 --> 00:36:26.160
 to give us the function f of t we calculated the Laplace transform of f of t which gave us f of s

284
00:36:26.160 --> 00:36:32.160
 so the Laplace transform of f t is f of s and then we had to do the final step wherever we saw an s

285
00:36:32.160 --> 00:36:41.840
 in f of s we had to shift it by the amount alpha we replaced the s by all occurrence of s in f of s

286
00:36:41.840 --> 00:36:50.080
 we replaced by s plus or minus alpha whatever the value of alpha was and that is summarized in number

287
00:36:50.080 --> 00:36:56.400
 22 in our table so I'll do an example this is an example from last time just to remind ourselves

288
00:36:56.400 --> 00:37:00.640
 of what was involved here today what we're actually going to do is we're going to use

289
00:37:01.200 --> 00:37:06.800
 the first shift in theorem to calculate inverse Laplace transforms and all that is this whole

290
00:37:06.800 --> 00:37:14.640
 procedure in reverse but let's recap what we did last time so here's an example in the time domain

291
00:37:14.640 --> 00:37:22.080
 we want the Laplace transform of this function we want to go to the s domain so we're going to use

292
00:37:22.080 --> 00:37:28.880
 the diagram here we've got our function e to the minus 4t t to the power six and I know that because

293
00:37:28.880 --> 00:37:35.680
 I'm multiplying the function t to the power six by an exponential I know that's the first

294
00:37:35.680 --> 00:37:43.040
 shift in theorem that I need to use so my diagram is here and I compare my starting function

295
00:37:43.040 --> 00:37:51.680
 e to the minus alpha t f of t with e to the minus 4t t to the six and from that I can easily

296
00:37:51.680 --> 00:37:58.400
 identify that alpha is equal to four just all it is is pattern matching so alpha must equal four

297
00:37:59.200 --> 00:38:05.200
 and f of t well f of t just to grow the exponential f of t is t to the power six

298
00:38:06.480 --> 00:38:13.440
 and I then go to the tables that my diagram tells me that I must now calculate the Laplace

299
00:38:13.440 --> 00:38:19.280
 transform of f of t to give me big f of s the tables would give me that with a little bit of work

300
00:38:19.920 --> 00:38:26.560
 so what have I got I've got t to the power six that's going to be number four with n equal to six

301
00:38:26.560 --> 00:38:34.400
 so it'll give me six factorial over s to the six plus one in other words six factorial over s to the

302
00:38:34.400 --> 00:38:45.120
 power seven six factorial 720 so I've got 720 over s to the power seven that's my f of s the final

303
00:38:45.120 --> 00:38:54.560
 step in my diagram tells me that I must replace every occurrence of s obtained at step two this

304
00:38:54.560 --> 00:39:02.400
 expression here I must replace every occurrence of s by s plus alpha alpha was equal to four so I've

305
00:39:02.400 --> 00:39:13.040
 got to replace s by s plus four and that gives me my Laplace transform of this function here e to the

306
00:39:13.040 --> 00:39:19.200
 minus four t t to the power six of this product involving an exponential function using the first

307
00:39:19.200 --> 00:39:24.480
 shift of theorem I obtain its Laplace transform so one thing I think I said last time that you've

308
00:39:24.480 --> 00:39:33.280
 got to be very careful about a common mistake when people get to the last step here they write down

309
00:39:33.280 --> 00:39:42.000
 that the Laplace transform was 720 over and they look at step two the big f of s and they write s to

310
00:39:42.000 --> 00:39:48.560
 the seven plus four that's not that's incorrect because what you're actually supposed to do is

311
00:39:48.560 --> 00:39:54.640
 you if s is raised to the power seven and you're replacing s by s to the power four then you must

312
00:39:54.640 --> 00:40:01.200
 raise the s to the power four to the power seven so make sure that you know the difference between

313
00:40:01.200 --> 00:40:10.800
 this and this because that's a fairly common mistake to make okay so that was what we did last week

314
00:40:10.880 --> 00:40:20.560
 and today we're going to apply this process in reverse so that when we're given a function

315
00:40:20.560 --> 00:40:27.520
 that's shifted in the s domain and you know it's shifted if you see that you've got an s

316
00:40:27.520 --> 00:40:34.960
 plus or minus some number in your big f of s when you get a function like that with a shift in the s

317
00:40:35.920 --> 00:40:43.440
 then you use the first shifted theorem in reverse and all it is is you do the steps backwards you go

318
00:40:43.440 --> 00:40:51.120
 around the diagram in the anticlockwise direction so first of all it tells you ignore the alpha

319
00:40:51.680 --> 00:41:00.480
 so I would do that so ignore the four out of 720 over s to the power seven and then I would go to get

320
00:41:00.480 --> 00:41:07.200
 by time domain function I would do the inverse Laplace of this one here and that would give me

321
00:41:07.200 --> 00:41:15.360
 t to the six and then the final step would tell me multiply t to the six by e to the minus alpha t

322
00:41:15.360 --> 00:41:23.040
 alpha we know is four so that would retrieve my time domain function so let's just have a look at

323
00:41:23.040 --> 00:41:29.600
 some examples of that before we do just we'll have a couple of words about it again

324
00:41:33.200 --> 00:41:38.160
 so this really just covers what I've been saying so calculate the inverse Laplace transform

325
00:41:38.160 --> 00:41:46.400
 of a functional form f of s plus alpha in other words f of s as in this case in step two that's

326
00:41:46.400 --> 00:41:57.040
 been shifted in the s domain because you've got s plus some number so this is it really this is what

327
00:41:57.040 --> 00:42:02.640
 we want to get we've got Laplace transform e to the minus alpha t f of t gives us a shifted function

328
00:42:02.640 --> 00:42:09.920
 in the s domain so we want to retrieve this function so we want the inverse Laplace of f of s plus alpha

329
00:42:09.920 --> 00:42:15.760
 to give us our time domain function where we've got some function of time multiplied by an

330
00:42:15.760 --> 00:42:21.760
 exponential so if I see a function like this with a shift in it I know it's going to be this type of

331
00:42:21.760 --> 00:42:28.800
 function some exponential multiplying some other time domain function and there's a video you can

332
00:42:28.800 --> 00:42:37.200
 watch here talking about the first shift in theorem being used in reverse so the procedure

333
00:42:38.000 --> 00:42:43.120
 as I've already mentioned is just what we did last week and in the example I just covered

334
00:42:43.760 --> 00:42:50.400
 but in the opposite direction going in reverse three steps we go around the diagram anti-clockwise

335
00:42:50.400 --> 00:42:57.360
 today rather than clockwise when we're going from the time domain to the s domain last time

336
00:42:58.240 --> 00:43:02.640
 so let's just go straight into an example and have a look at what's involved here to determine the

337
00:43:02.640 --> 00:43:12.160
 inverse Laplace transform of it would be f of s is equal to seven over s plus two to the four

338
00:43:13.280 --> 00:43:23.280
 so that would be sorry that would be my s f of s plus alpha so it's f of s plus two so I've seen the

339
00:43:23.280 --> 00:43:31.280
 shift so I'm pretty sure it's going to be the first shift in theorem in reverse so I go to my

340
00:43:31.280 --> 00:43:41.040
 diagram and what does that tell me to do well that tells me to ignore the alpha that will leave me f of s

341
00:43:42.000 --> 00:43:47.280
 so if I ignore the alpha what's the value of alpha here well it's whatever is added or subtracted

342
00:43:47.280 --> 00:43:54.720
 from the s in this case we've added a two so basically I ignore the two and that leaves me

343
00:43:55.040 --> 00:44:05.040
 seven over s to the power four so that's my f of s so up here I want to invert this one and to do that

344
00:44:05.040 --> 00:44:13.040
 I go to my table I look down this far right column here looking focusing on the denominator well I

345
00:44:13.040 --> 00:44:20.320
 know actually I don't have I stop at s cubed so s to the four I need to use this one with n equal to

346
00:44:20.400 --> 00:44:32.480
 three okay so I need n equal to three so I would have three factorial over s to the four if I was

347
00:44:32.480 --> 00:44:40.160
 going to use this one here that's what I've got to have if I've got s to the four in the denominator

348
00:44:40.160 --> 00:44:46.240
 that means n is equal to three so to give me three factorial over s to the four that's not exactly

349
00:44:46.240 --> 00:44:51.840
 what I've got three factorial is six but I've got a seven in the top line so I do the same as I did

350
00:44:51.840 --> 00:45:01.040
 with the earlier examples I write down in the top line exactly what the table would have me use and that

351
00:45:01.760 --> 00:45:13.040
 would be three factorial so on the outside I'm multiplied by this fraction whose top line is

352
00:45:13.040 --> 00:45:19.360
 the thing that I want which is a seven and on the bottom I've got what the table forced me to

353
00:45:20.000 --> 00:45:25.600
 use and that was three factorial and you can see the three factorial is cancelled to give me seven

354
00:45:25.600 --> 00:45:34.560
 over s to the four which is exactly where I've got so I can use this directly now the tables that's

355
00:45:34.560 --> 00:45:42.560
 just number four in the tables with n equal to three so that'll be t cubed and here three factorial is

356
00:45:43.120 --> 00:45:51.280
 seven over six times t cubed so that's it there so that's my f of t the last step says multiply

357
00:45:51.280 --> 00:45:59.120
 that f of t by e to the minus alpha t alpha we said was equal to two so I get seven over six

358
00:45:59.840 --> 00:46:07.680
 e to the minus two t times t cubed or if you prefer seven over six t cubed times e to the minus

359
00:46:08.560 --> 00:46:14.000
 the order there doesn't matter you know that's that's absolutely fine that's absolutely fine

360
00:46:14.880 --> 00:46:21.760
 okay so let's have a look at another one now we could use number 14 in the tables for this one

361
00:46:21.760 --> 00:46:29.680
 you'll find this or you know something similar to this in the tables but as I've said before

362
00:46:29.680 --> 00:46:36.240
 not every set of tables will give you that so really you need to know how to use the first

363
00:46:36.240 --> 00:46:44.400
 shifted theorem for this type of function so first of all how do I know it's the first

364
00:46:44.400 --> 00:46:51.280
 shift in theorem well I've got s plus or minus some number so that's a pretty good indication that

365
00:46:51.280 --> 00:47:02.080
 at least we should be trying the first shift in theorem anyway so what have I got so this is going

366
00:47:02.080 --> 00:47:10.880
 to be f of s plus alpha what's the alpha well it's minus two so it's f of s minus two so I want to

367
00:47:10.880 --> 00:47:19.760
 find f of s well that's easy I just ignore the alpha so it'll give me three over s squared plus nine

368
00:47:19.760 --> 00:47:25.760
 that's if I ignore the minus two I get three over s squared plus nine what's the value alpha well

369
00:47:25.760 --> 00:47:33.760
 it's minus two okay for now it's not two because in this setup it's s plus alpha we're using here

370
00:47:33.760 --> 00:47:40.640
 and we've got s minus two here so alpha must be minus two so we've got alpha we've got f of s

371
00:47:42.320 --> 00:47:52.240
 now there's f of s there so I want to um get the inverse Laplace of that term that's f of s to give

372
00:47:52.240 --> 00:48:00.960
 me f of t so the inverse Laplace of this one here if we looked in our tables we would find that this

373
00:48:00.960 --> 00:48:08.160
 will be number eight if I could go back what am I going I've got three over s squared plus nine

374
00:48:08.800 --> 00:48:17.200
 so if I've got a number eight anywhere here there should be one somewhere yeah I've I've got three

375
00:48:17.200 --> 00:48:24.080
 over s squared plus nine nine is three squared so what I've got is three over s squared plus three

376
00:48:24.080 --> 00:48:32.880
 squared so it's just sine three t so let's go back to my example there we are it's just sine

377
00:48:32.880 --> 00:48:38.880
 three t because I've got omega over s squared plus omega squared where omega is equal to three

378
00:48:39.680 --> 00:48:48.960
 so there we are so that's fine sine three t so that's f of t my last step says multiply the f of t

379
00:48:48.960 --> 00:48:57.280
 I got step two by e to the minus alpha t I've already identified that alpha was minus two so just be

380
00:48:57.280 --> 00:49:05.680
 careful with it it's going to be e to the minus minus two t in other words e to the two t so my answer

381
00:49:05.680 --> 00:49:15.040
 is e to the two t sine three t so that is the function of time that's the function f of t

382
00:49:15.040 --> 00:49:25.280
 use Laplace transform is given by this expression up here okay so that was the first shift in

383
00:49:25.280 --> 00:49:34.240
 theorem being used to calculate inverse Laplace transforms let's now move on and look at

384
00:49:35.920 --> 00:49:42.240
 some other functions that require algebraic manipulation before we can use the tables and

385
00:49:42.240 --> 00:49:49.760
 this will require us first of all the types of functions we'll look at will involve using partial

386
00:49:49.760 --> 00:49:57.120
 fractions so we'll have three different types of these and then to finish off with we look at

387
00:49:57.120 --> 00:50:05.600
 functions that involve us completing the square in the denominator mainly because the denominator

388
00:50:05.680 --> 00:50:12.080
 doesn't factorize easily to use partial fractions but I'll discuss that a little bit more shortly

389
00:50:12.080 --> 00:50:18.560
 so first of all we've got um partial fractions and as I said we're going to look at three

390
00:50:19.760 --> 00:50:26.480
 important different types of partial fractions all based on the type of function that's appearing in

391
00:50:26.480 --> 00:50:35.040
 the denominator of the function we want to find the inverse Laplace transform of so um in this case

392
00:50:36.080 --> 00:50:45.600
 here we are here's my function here now one way and the best way to go about this is first of all

393
00:50:45.600 --> 00:50:52.320
 let's see if that denominator is a quadratic function let's see if we can make things a

394
00:50:52.320 --> 00:51:00.000
 little easier for ourselves by factorizing this denominator well if you you know if you either

395
00:51:00.000 --> 00:51:07.040
 use trial and error or whatever you could find that this term here factorizes into two linear

396
00:51:07.040 --> 00:51:14.240
 terms a product of two linear terms s minus two s plus one and just checking it s times s is s squared

397
00:51:15.680 --> 00:51:22.480
 s times one is s minus two s gives me minus s minus two times one is minus two so that's fine

398
00:51:22.480 --> 00:51:30.320
 that's the correct factorization for that expression in the denominator now most of you

399
00:51:30.320 --> 00:51:36.800
 will have seen partial fractions before but I'll go into a little bit of detail as we're working

400
00:51:36.800 --> 00:51:43.760
 through them so in this case in the denominator we've been able to factorize the quadratic into two

401
00:51:43.760 --> 00:51:51.200
 distinct linear terms distinct just mean that they're different s minus two times s plus one so

402
00:51:51.200 --> 00:51:57.360
 they're two different two distinct linear terms so what I'm going to need is I'm going to need a

403
00:51:57.360 --> 00:52:08.080
 fraction for each term and with partial fractions I'm going to break this into two separate fractions

404
00:52:08.080 --> 00:52:16.720
 I'm going to have a over s minus two plus b over s plus one so a fraction for each term now with

405
00:52:16.720 --> 00:52:24.000
 partial fractions the numerator always has degree one less than the denominator now I said these are

406
00:52:24.000 --> 00:52:32.160
 two distinct linear terms linear meaning that the highest power of s is one so they're both

407
00:52:32.160 --> 00:52:39.120
 s to the power one so the numerator will have degree one less so it'll be s to the zero in other

408
00:52:39.120 --> 00:52:45.920
 words s to zero is one so that's why you have a constant in the top line because you get a linear

409
00:52:45.920 --> 00:52:54.320
 factor in the denominator so that means a constant goes in the top line so that's our fractions and

410
00:52:54.320 --> 00:53:01.120
 we need to find a and b and then once we've broke once we've identified a and b these will become

411
00:53:01.120 --> 00:53:11.120
 quite easy because if you remember there'd be just be a number over s plus a number and that's

412
00:53:11.120 --> 00:53:17.360
 very like you know tables one over s plus alpha number five in our tables but let's look at a

413
00:53:17.360 --> 00:53:23.360
 little bit of work to do before we finally get to a and b so first of all the first thing to do here

414
00:53:23.920 --> 00:53:31.920
 is I want to get an equation where I can find a and b so I'm going to multiply both sides of this

415
00:53:33.200 --> 00:53:40.400
 equation here by the denominator of the left hand side so I multiply this by its own denominator I

416
00:53:40.400 --> 00:53:50.000
 get five s minus one move it over just if I get that multiply the first fraction here the s minus

417
00:53:50.000 --> 00:53:58.640
 two is cancelled to give me a into s plus one and then move it over to here and you can see the s

418
00:53:58.640 --> 00:54:07.200
 plus one's cancelled to give me b into s minus two okay so once we get to this stage we've got one

419
00:54:07.200 --> 00:54:15.120
 equation with two unknowns which might seem a bit of a problem but we can set values of s such

420
00:54:15.120 --> 00:54:23.520
 that we end up with one equation with one unknown and basically you just look at what will make

421
00:54:23.520 --> 00:54:31.520
 either bracket equal to zero so first of all I'm going to take s equal to minus one if I set s equal

422
00:54:31.520 --> 00:54:41.040
 to minus one I'll have a times minus one plus one or a times zero so that'll vanish and s minus one

423
00:54:41.040 --> 00:54:48.560
 I'll have b times minus one minus two minus three so I minus three b there s minus one here I'll have

424
00:54:48.560 --> 00:54:55.280
 minus five minus one which is minus six so I'll have minus six is equal to well that's zero

425
00:54:56.240 --> 00:55:04.240
 minus three b in other words b is equal to two okay so I'm done with that one um

426
00:55:05.280 --> 00:55:11.040
 any other values that I can set s equal to two to make something vanish well yes I can set s equal

427
00:55:11.040 --> 00:55:18.240
 to two and I'll set s equal to two this then vanishes and just leave so in equation five times two is

428
00:55:18.240 --> 00:55:28.560
 ten minus one is nine there's the nine there s is equal to two i'm three a nine is equal to three a

429
00:55:28.560 --> 00:55:34.080
 that part is zero so I just got three a is equal to nine or a is equal to three so the partial

430
00:55:34.080 --> 00:55:39.600
 fractional representation of this term here the thing that I started out with is given at the

431
00:55:39.600 --> 00:55:47.360
 foot of the slide here okay so that now just leaves me to find the inverse Laplace transform

432
00:55:47.440 --> 00:55:55.440
 of these two individual fractional terms I don't have anything that looks like that in my tables

433
00:55:55.440 --> 00:56:02.800
 don't have anything that looks remotely like it but partial fractions have allowed me to obtain

434
00:56:02.800 --> 00:56:10.160
 two terms that I can easily find the inverse Laplace transform of so here I am so there's my

435
00:56:10.160 --> 00:56:15.040
 original expression that I want the inverse Laplace transform of I broke it up into its partial

436
00:56:15.520 --> 00:56:23.360
 representation by calculating these two fractions the linearity property says that I can just take

437
00:56:23.360 --> 00:56:29.440
 the inverse Laplace transform of each one individually and add them together any constants I can take

438
00:56:29.440 --> 00:56:36.640
 outside that's all down to the linearity property I've got one over s minus two look at the bottom

439
00:56:36.640 --> 00:56:44.640
 line and I hit this one s plus a constant well you can take it with a constant b minus two it's

440
00:56:44.720 --> 00:56:50.960
 s plus minus two so that's just going to give me that alpha is minus two so this becomes e to the

441
00:56:50.960 --> 00:56:58.720
 minus minus two t for e to the two t for this part here don't forget though it's multiplied by three

442
00:56:58.720 --> 00:57:06.560
 and for this one I've got one over s plus alpha one over s plus one that again it's number five

443
00:57:06.560 --> 00:57:12.560
 but this time with alpha equal to one and i've e to the minus one t don't forget there's a two

444
00:57:12.560 --> 00:57:23.280
 outside there so I get two e to the minus t so that's um that's the answer to that one so that

445
00:57:23.280 --> 00:57:33.760
 was the case when we were given a function whose denominator was a quadratic who were able that

446
00:57:33.760 --> 00:57:40.640
 were able to factorize quite easily into the product of two linear terms and so we then

447
00:57:41.360 --> 00:57:47.520
 got two fractions one for each term in the denominator and with a little bit of algebra

448
00:57:47.520 --> 00:57:53.040
 we were able to solve for the unknown constants a and b and that left us a fairly straightforward

449
00:57:53.040 --> 00:57:59.840
 task using number five in the tables to find the inverse Laplace transform of our original function

450
00:57:59.840 --> 00:58:08.480
 as given here and I think I've mentioned before that number five will be seeing quite a lot of this

451
00:58:09.040 --> 00:58:16.160
 this one because the solutions to differential equations tend to involve exponential functions

452
00:58:16.160 --> 00:58:22.640
 so number five maybe number six seven and so on they tend to feature quite a lot that's the ones

453
00:58:22.640 --> 00:58:29.040
 that involve exponential functions so it's good to be familiar exactly with with these ones with

454
00:58:29.040 --> 00:58:36.560
 the exponential involving the exponentials okay moving on now here's another one and this is a

455
00:58:36.560 --> 00:58:45.760
 different type of partial fraction again you know you would look at this and you might be tempted to

456
00:58:45.760 --> 00:58:56.240
 say well I could write that that fraction there that denominator I could write it as the product

457
00:58:56.880 --> 00:59:03.120
 of three linear terms because that's s to the one s to the one s minus six and I could just

458
00:59:03.120 --> 00:59:10.240
 get my partial fraction a over s plus b over s plus c over s minus six that's one fraction

459
00:59:10.240 --> 00:59:17.440
 for each term but be very careful that will not work this is what we call this is a repeated

460
00:59:17.440 --> 00:59:24.800
 linear factor we've got s times s it's a repeated linear factor and using this formulation will not

461
00:59:24.800 --> 00:59:32.720
 work we'll end up with total nonsense when we try and get to this stage here when we're trying to

462
00:59:32.720 --> 00:59:42.320
 simplify it so how do we get around that well the way we get around that is we set up a partial

463
00:59:42.320 --> 00:59:48.160
 fraction square one fraction for the s minus six term that's fine there it is we've got one fraction

464
00:59:48.160 --> 00:59:56.800
 for s b over s and rather than writing another fraction like a c over s we write it as c over s

465
00:59:56.800 --> 01:00:08.080
 squared okay so that's what we do we write it like that okay so uh let's try and get an equation

466
01:00:08.080 --> 01:00:13.680
 that we can work with out of that well again we do the same as we did in the last example we multiply

467
01:00:13.680 --> 01:00:20.960
 both sides by the denominator of the left hand side and as you can see here multiplying this first

468
01:00:20.960 --> 01:00:28.240
 term by this denominator which I just copied over here in red the s minus six cancel to give me a

469
01:00:28.240 --> 01:00:37.680
 times s squared moving this on to the next term on the right hand side I multiply this this term by

470
01:00:37.680 --> 01:00:45.680
 the denominator s squared into s minus six that s will cancel one of them giving me b times s into

471
01:00:45.680 --> 01:00:52.800
 s to the minus six that second term and finally I multiply my third fraction the right hand side

472
01:00:52.800 --> 01:01:00.480
 by the denominator of the left hand side the s squared this time cancel to give me c into s minus

473
01:01:00.480 --> 01:01:08.800
 six that's it there okay so we've now got one equation with three unknowns which on the face of

474
01:01:08.800 --> 01:01:18.160
 it would almost seem impossible to solve but again we can set values of s so that some of these

475
01:01:18.160 --> 01:01:27.040
 unknowns disappear and leave us simple equations to solve so for example let's choose a value of s

476
01:01:27.040 --> 01:01:34.960
 that will eliminate some of these terms so clearly if I said s equal to zero that'll vanish with s

477
01:01:34.960 --> 01:01:43.280
 equal to zero that will also vanish and this will just be minus six c there's no s there so that just

478
01:01:43.280 --> 01:01:54.000
 stays as 18 so I'll get 18 is equal to minus six c that's with s equal to zero in this equation here

479
01:01:55.360 --> 01:02:02.400
 that vanishes that vanishes and I've just got c times zero minus six or minus six c and that's

480
01:02:03.120 --> 01:02:09.840
 18 on the right hand so c is equal to minus three okay that's fine can anything else be

481
01:02:09.840 --> 01:02:18.400
 eliminated well yes with setting s equal to six then that term will completely go because this bracket

482
01:02:18.400 --> 01:02:24.320
 will be six minus six which is zero so that'll all go so to you will this because you get six

483
01:02:24.320 --> 01:02:32.160
 minus six times c so it's c times zero so that'll completely vanish so if I just get my tablet uh

484
01:02:32.160 --> 01:02:38.080
 what am I got I've got that this term has gone and I've got that this term has gone

485
01:02:38.800 --> 01:02:53.200
 s was equal to six so I've got 18 is equal to 36 a so in other words a must be equal to 18 over 36

486
01:02:53.280 --> 01:03:03.040
 or a is equal to a half so that's okay that's that one solved now can I eliminate anything else

487
01:03:03.040 --> 01:03:08.880
 any other value of s that would help me well not really because I've said s equal to zero

488
01:03:08.880 --> 01:03:14.160
 and I've said s equal to six and that's the only things that could directly eliminate any of these

489
01:03:14.960 --> 01:03:22.560
 three expressions and I've still left to find b I found c and I found a I've still got to find b

490
01:03:23.360 --> 01:03:30.960
 but the way you do that is you just choose any value for s that you haven't used before

491
01:03:30.960 --> 01:03:37.920
 and try and keep it as a simple value we've used zero so we can't use that we use six we can't use

492
01:03:37.920 --> 01:03:46.640
 that either so let's just choose the most sensible value and that's s equal to one so let's say s

493
01:03:46.640 --> 01:03:53.920
 equal to one here well that still gives me 18 on the left hand side so that'll be 18 there we are

494
01:03:53.920 --> 01:04:02.480
 there it is there and with s equal to one that'll just give me a s equal to one just be careful b

495
01:04:02.480 --> 01:04:10.480
 times one times one minus five so that'll give me minus five b there it is there and again I'll have

496
01:04:10.480 --> 01:04:20.800
 one minus six remember s is equal to one so I'll have minus five c now I know what a is a half

497
01:04:23.200 --> 01:04:34.160
 c is minus three so I can work all of this out I can 18 is equal to a half minus five b plus 15

498
01:04:34.160 --> 01:04:43.040
 if I simplify this a little and solve it for b I find that b is equal to minus a half so I've

499
01:04:43.040 --> 01:04:51.120
 now been able to find a b and c I've been able to find a b and c and I can substitute them back in here

500
01:04:52.080 --> 01:04:59.840
 so that's what I'm going to do on the next slide so if I just copy my expression over

501
01:04:59.920 --> 01:05:06.240
 so let's see exactly what we've done so just down here I've got my expression there's the left hand

502
01:05:06.240 --> 01:05:16.160
 side that's it up there I had that a was equal to a half so I get one over two into s minus six there

503
01:05:16.160 --> 01:05:27.680
 it is I had that b was equal to minus a half so what do I have b is minus a half so I get

504
01:05:27.680 --> 01:05:35.920
 minus one over two s there it is there and I had that c was equal to minus three so I get minus three

505
01:05:35.920 --> 01:05:42.160
 over s squared there it is there so that's just substituting the a b and c values I found in the

506
01:05:42.160 --> 01:05:51.680
 previous slide into my general partial fraction expression here and I've obtained this expression

507
01:05:51.680 --> 01:06:00.880
 up here so now I can try and use my tables to find the inverse Laplace transform so the

508
01:06:00.880 --> 01:06:08.480
 inverse Laplace transform of my original function 18 over s squared into s minus six is equal to

509
01:06:09.120 --> 01:06:15.840
 the inverse Laplace transform of all of that but by the linearity property I can just take

510
01:06:15.840 --> 01:06:24.080
 inverse Laplace of each individual term and then do my subtractions right so this one first of all

511
01:06:24.080 --> 01:06:31.280
 well I think we know this one by now it's just the two we can ignore that for now to look after

512
01:06:31.280 --> 01:06:38.880
 itself it's just one over s plus alpha that's number five so that's all I've done here is I've

513
01:06:38.880 --> 01:06:44.560
 taken the one over two outside the operator outside the inverse Laplace operator to give me

514
01:06:45.120 --> 01:06:51.520
 this and I've just said it's number five with alpha equal to minus six so it'll be e to the

515
01:06:51.520 --> 01:06:58.960
 60 don't forget multiplied by a half here I've thought of this as minus one over minus a half

516
01:06:58.960 --> 01:07:05.040
 times one over s so I've taken the minus a half outside one over s the inverse Laplace of that's

517
01:07:05.040 --> 01:07:15.680
 just one so I get minus a half this one I've just taken the three outside and I've got minus three

518
01:07:15.680 --> 01:07:22.240
 inverse Laplace of one over s squared which is just number two which is t so I get minus three t

519
01:07:22.240 --> 01:07:28.400
 put it all together and that is the inverse Laplace transform of my original function there it is

520
01:07:28.400 --> 01:07:38.560
 given here okay so that's the second type of partial fraction we're looking at this was different

521
01:07:38.560 --> 01:07:45.040
 from the first one the first one we had distinct linear terms in the denominator in this case

522
01:07:45.040 --> 01:07:52.560
 we've got a repeated linear term in the denominator so we had to treat it slightly different when we

523
01:07:52.640 --> 01:07:59.280
 set up our partial fraction representation we had to take account of the repeated linear factor

524
01:07:59.280 --> 01:08:06.240
 and we did that by including the c over s squared term rather than having

525
01:08:08.560 --> 01:08:15.520
 plus b over s plus c over s I had to have an s squared here right okay so the third

526
01:08:16.480 --> 01:08:25.040
 the third case that we're going to look at is when we've got a quadratic factor in the denominator

527
01:08:26.320 --> 01:08:32.720
 that's what's known as irreducible basically you cannot factorize it into two linear terms

528
01:08:32.720 --> 01:08:39.040
 without using complex numbers it does not factorize easily you know and you can try and factorize this

529
01:08:39.040 --> 01:08:45.280
 but but you won't be able to do that without using complex numbers it does actually factorize to become

530
01:08:45.600 --> 01:08:52.480
 s plus 2j times s minus 2j but we don't want to be using that with partial fractions or our Laplace

531
01:08:52.480 --> 01:09:01.520
 tables we want this you know expressed in terms that are given in our tables or similar to our

532
01:09:01.520 --> 01:09:09.600
 terms in our tables so so we've got to consider this as the product in the denominator of a quadratic

533
01:09:09.600 --> 01:09:16.800
 because we've got an s squared and a linear term well we've seen linear terms before and we know it's

534
01:09:16.800 --> 01:09:23.920
 just going to be a constant over whatever the linear term is so you know that's where that comes

535
01:09:23.920 --> 01:09:30.800
 from c over s minus 4 what about the quadratic term well if you remember earlier on I said that with

536
01:09:30.800 --> 01:09:38.480
 partial fractions the order of the numerator is always one less than the denominator so you could

537
01:09:38.480 --> 01:09:47.920
 a linear term a power one term in the denominator then you need a power zero term in the top line

538
01:09:47.920 --> 01:09:56.400
 the order of the numerator in terms of s is always one less than the denominator so an s to the one

539
01:09:56.400 --> 01:10:02.720
 down below means you get an s to the zero on top s to the zero is just one so that's why you get a

540
01:10:02.720 --> 01:10:11.280
 constant up top now this other term is not a linear term it's a quadratic term the denominator so

541
01:10:11.280 --> 01:10:18.720
 that means if the function that goes up top has degree one less this I've ordered one less than

542
01:10:18.720 --> 01:10:25.680
 this then it must be an s to the power one term a linear term and the most general linear function

543
01:10:26.320 --> 01:10:36.160
 in s would be as plus b so um that's what we're going to use that's what we're going to use so

544
01:10:36.160 --> 01:10:42.720
 there are two terms one this one for the linear term and this one for the quadratic term that appears

545
01:10:42.720 --> 01:10:50.960
 in the denominator again we multiply both sides by the denominator left hand side so you multiply

546
01:10:50.960 --> 01:11:00.880
 that by its own denominator just gives me three s squared plus 28 move it over now to multiply

547
01:11:01.920 --> 01:11:09.040
 the first term on the right hand side the s squared plus four cancels with that s squared plus four

548
01:11:09.040 --> 01:11:15.440
 it gives me as plus b times s minus four so that's where that comes from and then finally

549
01:11:16.320 --> 01:11:21.360
 I multiply the second term on the right hand side by the denominator of the left hand side

550
01:11:22.320 --> 01:11:31.120
 and the s minus four cancels to give me c times s squared plus four so that gives me one equation

551
01:11:31.840 --> 01:11:41.840
 this time again with three unknowns a b and c which I've got to find and again I try and choose my s

552
01:11:41.840 --> 01:11:49.200
 values wisely so that I can make this equation simpler by eliminating terms is there anything that I

553
01:11:49.200 --> 01:11:55.520
 can do here to eliminate terms well I'm looking looking at the right hand side of course yes I can

554
01:11:55.520 --> 01:12:05.920
 set s equal to four and that will eliminate the a and the b because I'll have as plus b times four

555
01:12:05.920 --> 01:12:13.520
 minus four that's where s equal to four four minus four is zero so that'll all go so if I said s

556
01:12:13.520 --> 01:12:20.320
 equal to four on the left hand side I'll have three times 16 is 48 minus 28 that gives me that's where

557
01:12:20.320 --> 01:12:30.240
 the 20 there comes from that has we've just said zero four squared 16 plus four is 20 so I'll have

558
01:12:30.240 --> 01:12:40.720
 20 is equal to 20 c in other words c is equal to one okay so move on can we choose another value for s

559
01:12:40.720 --> 01:12:49.840
 that will make something disappear or vanish well no there isn't actually because you know we've got

560
01:12:49.840 --> 01:12:57.600
 nothing we can't set a value here s squared plus four equals zero well you need to have a complex

561
01:12:57.680 --> 01:13:03.760
 number it'd be either 2g or minus 2g that that's not going to work for us that's not going to work for us

562
01:13:06.160 --> 01:13:14.000
 so the thing to do here is you choose a value for s that you haven't used before

563
01:13:15.520 --> 01:13:21.280
 you choose a value for s that we haven't chosen before and make it a sensible value so I'm going

564
01:13:21.280 --> 01:13:27.920
 to say that s equal to zero so if I do that my left hand side becomes minus 28 there it is

565
01:13:29.200 --> 01:13:37.120
 my right hand side with s equal to zero well the a will actually go here and here i'll have minus four

566
01:13:37.120 --> 01:13:45.120
 so I just have minus four b to this part and here i'll have four c so i've got minus 28 is equal to

567
01:13:45.120 --> 01:13:54.320
 minus 4b plus 4c there's my equation there it is now I know what c is c is actually equal to one

568
01:13:55.520 --> 01:14:05.760
 so that'll be that'll become four moving you over I get minus 32 is equal to minus 4b

569
01:14:06.320 --> 01:14:16.160
 or minus 4b is equal to minus 32 or b is equal to eight so that is the solution for b so I've got

570
01:14:16.160 --> 01:14:23.440
 c and I've got b now I need to find a so again I choose a sensible value for s that I haven't used

571
01:14:23.440 --> 01:14:32.640
 before so I'm just going to set s equal to one so my left hand side becomes minus 25 three times

572
01:14:32.640 --> 01:14:43.200
 one minus 28 three minus 28 minus 25 there it is with s equal to one I'll get a plus b times minus

573
01:14:43.200 --> 01:14:54.160
 three so I'll have minus 3a minus 3b s equal to one will give me plus 5c so there's my equation down

574
01:14:54.160 --> 01:15:05.440
 here I know what c is and I know what b is and that will allow me to find a and when I do that

575
01:15:05.440 --> 01:15:11.360
 I'm going to substitute the b and the c values into this equation here fairly easy I find that a is

576
01:15:11.360 --> 01:15:18.560
 equal to two so I've got a b and c and I can substitute them back into my partial fracture

577
01:15:18.560 --> 01:15:26.800
 representation up here and I obtain the expression at the foot of the slide so this is the thing that

578
01:15:26.800 --> 01:15:33.120
 I'm going to now try and find the partial fracture representation of I'm going to try and find the

579
01:15:33.120 --> 01:15:38.400
 in-virtual-a-plus transform of this is the partial fracture representation of this original

580
01:15:38.400 --> 01:15:46.000
 expression which no matter how much I looked in the tables I would never find an expression that

581
01:15:46.160 --> 01:15:55.760
 looks like this but I hope to find some terms that look like the partial fraction representation

582
01:15:55.760 --> 01:16:03.040
 that I've got so if I move over to the next slide I've got this thing here now this is an easy one

583
01:16:03.040 --> 01:16:12.080
 this again is number five one over s plus alpha that one there in this case alpha minus four so it'd

584
01:16:12.080 --> 01:16:19.920
 be e to the four t so that one's easy to deal with this one isn't quite as easy s squared plus four

585
01:16:19.920 --> 01:16:26.960
 well I would search my table it's s squared plus four just be careful it's not this one

586
01:16:26.960 --> 01:16:35.360
 because this this is s plus a number all squared whereas this one is just s squared plus a number

587
01:16:35.360 --> 01:16:43.120
 so sometimes people do actually choose this one but it's different that is different this is

588
01:16:43.120 --> 01:16:50.960
 the new denominator everything is squared in it here only the s squared so I would keep looking

589
01:16:50.960 --> 01:17:02.240
 keep looking and I would eventually hit number eight and nine s squared plus some number that's

590
01:17:02.240 --> 01:17:09.040
 called omega squared s squared plus omega squared now number eight has that the denominator so two

591
01:17:09.040 --> 01:17:17.120
 actually does number nine so the decision is which one of these am I going to use well I look at the

592
01:17:17.120 --> 01:17:25.520
 numerator and again unfortunately my numerator has more terms in it it doesn't exactly match either

593
01:17:26.080 --> 01:17:33.440
 but I can actually be a bit clever with this one and it's perfectly okay to split up the

594
01:17:33.440 --> 01:17:41.840
 numerator and have this thing as a sum of two fractions two s over s squared plus four plus

595
01:17:41.840 --> 01:17:52.320
 eight over s squared plus four okay so this should help me now I will now go and just check these

596
01:17:52.320 --> 01:17:58.720
 with the tables I know let's have a look at this one first of all s squared plus a number so it's

597
01:17:58.720 --> 01:18:05.040
 either going to be number eight or nine we've certainly decided on that what's in the top line

598
01:18:06.320 --> 01:18:13.520
 well this one the Laplace variable s appears in the top line here so that will tell me that it should

599
01:18:14.560 --> 01:18:20.480
 be number nine it's number nine not number eight because number eight only has a number

600
01:18:20.480 --> 01:18:27.280
 there is no s in the numerator in number eight so it's definitely number nine and I can take the

601
01:18:27.280 --> 01:18:34.000
 two outside if I like so I've just got to get this so that's pretty easy that's just number nine

602
01:18:34.560 --> 01:18:41.600
 with omega equal to two so it's just cos two t and don't forget the two on the outside so I've got

603
01:18:41.600 --> 01:18:47.920
 that one I did this one earlier on you know just one over s minus four so it's number five with alpha

604
01:18:47.920 --> 01:18:54.640
 equal to minus four so it's e to the minus minus four t it's just e to the four t this one

605
01:18:55.520 --> 01:19:02.640
 this one here I need to do something with this one well it's s squared plus four so we knew from

606
01:19:02.640 --> 01:19:09.280
 earlier this was going to be number eight or number nine again I examined the numerator now to decide

607
01:19:09.280 --> 01:19:15.120
 which one of those two candidates I should use well we know number nine has still a plus variable

608
01:19:15.120 --> 01:19:20.640
 in the top line and this one doesn't this one only has a number a constant so it's definitely number

609
01:19:20.640 --> 01:19:31.920
 eight but we have to be careful the omega down below here is two because it's two squared it's

610
01:19:31.920 --> 01:19:39.760
 omega squared so two squared is four so omega is equal to two so I must have a two in the top line

611
01:19:39.920 --> 01:19:47.360
 so there's the two there so you know I could write this I could have my fraction outside here

612
01:19:48.000 --> 01:19:53.360
 eight the thing that I want to divide by what the table want me to have which was two so that would

613
01:19:53.360 --> 01:19:59.840
 give me the four so I kind of jumped a step there so but that's the four there so I can then easily

614
01:19:59.840 --> 01:20:07.360
 invert this to give me sine two t because omega is equal to two and that's become four sine two t

615
01:20:07.360 --> 01:20:14.720
 and there is the investor class transform of that function that I started out with which looked

616
01:20:14.720 --> 01:20:18.800
 pretty hopeless really at the beginning because there was nothing in the table that looked

617
01:20:18.800 --> 01:20:28.720
 remotely like that but partial fractions allowed me to write this expression in a form that I could

618
01:20:28.800 --> 01:20:38.880
 use the tables with to invert okay so that's fine now in all these cases and most these

619
01:20:38.880 --> 01:20:47.680
 kids were able to factorize the denominator if we needed to there are some cases though

620
01:20:47.680 --> 01:20:55.040
 where we're unable to factorize the denominator easily without complex numbers or whatever we

621
01:20:55.040 --> 01:21:02.560
 really don't want to do that so the way we go about it about these is we complete the square

622
01:21:03.200 --> 01:21:07.520
 now we could very easily we could just as easily have complete the square in these

623
01:21:08.560 --> 01:21:14.880
 when we're able to factorize we can either factorize or complete the square either method will work

624
01:21:16.240 --> 01:21:23.280
 but if they don't factorize easily that avenue isn't available for us so all we can do is complete

625
01:21:23.280 --> 01:21:29.840
 the square so let's have a look at completing the square so there's our expression here so

626
01:21:31.120 --> 01:21:38.160
 you know how do I know this thing won't break up into two brackets s plus or minus something times

627
01:21:38.160 --> 01:21:47.200
 s plus or minus something well if I try to solve this thing equal to zero I might use the quadratic

628
01:21:47.200 --> 01:21:53.040
 formula to find the value of s and if I did that my what's known as the discriminant the quantity

629
01:21:53.040 --> 01:22:00.080
 under the square root sign in the quadratic formula b squared minus 4ac as you can see in this case

630
01:22:00.080 --> 01:22:07.040
 it works out to be negative 64 so I'm taking the square root of a negative number so that brings in

631
01:22:07.840 --> 01:22:14.640
 imaginary numbers and brings in complex numbers of course so this will not factorize easily

632
01:22:14.640 --> 01:22:21.120
 you will not get this s plus or minus an integer times s plus or minus an integer like we saw

633
01:22:21.120 --> 01:22:28.560
 previously that will not work so what we do is we complete the square here so we've got s squared

634
01:22:28.560 --> 01:22:36.880
 minus 6s plus 25 to complete the square who we do is we take the s here we take half of the

635
01:22:36.880 --> 01:22:43.920
 coefficient which is three from minus s we copy the minus down we have that all squared we subtract

636
01:22:43.920 --> 01:22:52.880
 the term we've got in here subtract the three square it don't forget the plus 25 now I'm sure

637
01:22:52.880 --> 01:22:58.080
 everyone had seen this before so I won't go into too much detail really on because it's in some of

638
01:22:58.080 --> 01:23:08.720
 the earlier maths courses so when I simplify this I get s minus three or squared plus 16 okay now one

639
01:23:08.720 --> 01:23:15.840
 thing to be careful here of here is this minus this minus three squared this is this up here shows

640
01:23:15.840 --> 01:23:23.440
 this is complete these two things minus three squared like we've got is minus three times three

641
01:23:23.440 --> 01:23:30.720
 which is minus nine it's completely different from minus three all squared which is minus three times

642
01:23:30.720 --> 01:23:38.080
 minus three which gives positive nine so make sure you know the distinction between these two things

643
01:23:38.080 --> 01:23:45.200
 this is subtract three squared in other words minus three times three and that's why

644
01:23:45.200 --> 01:23:53.040
 that would be minus nine plus 25 and you get 16 here so that's us complete the square so this

645
01:23:53.040 --> 01:24:00.080
 original expression here we can factorize into this expression like this we can then go to our tables

646
01:24:00.480 --> 01:24:10.160
 and you know we would look down and this certainly you know here you can think of this as s plus alpha

647
01:24:10.160 --> 01:24:18.000
 plus omega squared with alpha equal to minus three and omega equal to four so it's going to be one of

648
01:24:18.000 --> 01:24:25.120
 those two the first one though number 14 won't work because that's just a constant so this one

649
01:24:25.760 --> 01:24:30.560
 just similar to what we saw earlier where the cosine has a constant in the top line the sign

650
01:24:30.560 --> 01:24:37.360
 just has a constant it's going to be the cosine so it's going to be the cosine with alpha equal to

651
01:24:37.360 --> 01:24:45.600
 minus three see i've got s plus alpha top and bottom these must match these things must match

652
01:24:45.600 --> 01:24:52.800
 each other and they do i've got s minus three in both places here so alpha is equal to minus three

653
01:24:53.440 --> 01:25:00.400
 and omega which omega squared four square so omega is equal to four so i can easily write this as

654
01:25:01.200 --> 01:25:11.280
 e to the minus minus three t or e to the three t cause omega t becomes cos four t so that's the

655
01:25:11.280 --> 01:25:17.520
 initial plus transform of this expression here and again you can look at the table and you will

656
01:25:17.520 --> 01:25:23.920
 not find something that looks like this so you've got you've got no choice but to complete the square

657
01:25:23.920 --> 01:25:32.320
 the denominator does not factorize easily if it did we could use partial fractions but it doesn't

658
01:25:32.320 --> 01:25:37.600
 so we complete the square in the denominator move on to another example

659
01:25:41.040 --> 01:25:47.200
 maybe i'll write out the complete the square here i know i've done it on you can see it on screen there

660
01:25:47.200 --> 01:25:57.360
 but let's just see if i can maybe write it out let's get the pen going so i've got s squared plus

661
01:25:57.360 --> 01:26:05.760
 two s plus five so the rule for complete the square is i have my s there i have a square there

662
01:26:05.760 --> 01:26:12.080
 i'll subtract something and i've got plus five so what goes in here well that's a plus so i'll

663
01:26:12.080 --> 01:26:19.520
 copy that down half of that gets copied so there it goes and then whatever i write in here i subtract

664
01:26:19.520 --> 01:26:32.000
 it squared so this becomes s plus one all squared minus one plus five gives me plus

665
01:26:32.080 --> 01:26:43.600
 four like that which of course is equal to s plus one all squared plus two squared and i only write it

666
01:26:43.600 --> 01:26:50.160
 like that because i know that's the form that's needed for the tables so let's just compare them

667
01:26:50.960 --> 01:27:00.480
 so i look at the denominator i've got s plus one all squared plus two squared s plus one all squared

668
01:27:00.480 --> 01:27:06.720
 plus two squared so it's either going to be 14 or 15 let's now look at the numerator to narrow the

669
01:27:06.720 --> 01:27:12.640
 field down even further well the numerator is just a number so it's going to be number 14

670
01:27:13.760 --> 01:27:21.600
 15 won't work because 15 has the Laplace variable in there it's got the s in the numerator 14 is

671
01:27:21.600 --> 01:27:28.560
 perfect because it's just got a number but if you look at this carefully we don't quite have this in

672
01:27:28.560 --> 01:27:36.880
 the same form because the table tells me that whatever the omega here is and i must have exactly

673
01:27:36.880 --> 01:27:45.040
 the same omega value up here so i've got a two down here so that means i must have a two in the

674
01:27:45.040 --> 01:27:53.360
 numerator because look there's omega is equal to two and i'll have two squared there which is actually

675
01:27:53.360 --> 01:27:59.520
 what i've got so i need a two in the top line and that's all i've done here so this in here is

676
01:27:59.520 --> 01:28:06.800
 exactly what the table has me used so basically you know if you look at it i just clear this first

677
01:28:06.800 --> 01:28:17.520
 of all if you look at this this is of the form omega in the top line over s plus alpha squared

678
01:28:18.080 --> 01:28:23.680
 plus omega squared that's exactly in the form that i want to use number 14

679
01:28:24.960 --> 01:28:32.560
 so i've introduced a two in the top line to enable me to use number 14 but i must correct that because

680
01:28:32.560 --> 01:28:38.000
 that's not what i started out with and we know how to do that you multiply by a fraction this top

681
01:28:38.000 --> 01:28:43.280
 line has the thing that you want which is a one remember it's a one that we want in the top

682
01:28:43.360 --> 01:28:51.440
 i would divide by whatever the table forced me to introduce and that was a two i had to introduce

683
01:28:51.440 --> 01:29:00.720
 a two here because that omega had to match the omega down below so i multiply this by by a half

684
01:29:01.360 --> 01:29:07.600
 and then i can easily invert this bit because it's just number 14 with alpha equal to one

685
01:29:08.480 --> 01:29:16.720
 and omega equal to two so it gives me remembering the half it gives me a half e to the minus t

686
01:29:17.680 --> 01:29:25.840
 sine 2t it's just that one with alpha equal to one give me e to the minus one t omega equal to

687
01:29:25.840 --> 01:29:34.800
 sine 2t okay so another example of completing the square so what do we got so i think i'll

688
01:29:34.800 --> 01:29:45.200
 just write this one out as well so we've got s squared s squared minus six s plus 25 i can

689
01:29:45.200 --> 01:29:52.080
 break this into my brackets there's an s there there's a square there and a minus something

690
01:29:52.080 --> 01:29:58.640
 squared and don't forget the plus 25 there's a plus here so that's got to be a plus take half

691
01:29:58.640 --> 01:30:04.000
 of that coefficient so it gives me a three subtract the thing i've written in there so it's three

692
01:30:04.000 --> 01:30:15.280
 squared if you work that out you get s plus three all squared minus nine plus 25 so it's plus 16

693
01:30:15.280 --> 01:30:24.240
 or in other words s plus three all squared plus four squared there's a plus there okay

694
01:30:25.680 --> 01:30:32.560
 so that's exactly what i've got here okay i've just completed the square in the denominator

695
01:30:32.560 --> 01:30:37.520
 that's all i've done that's just this calculation going from left to right here that's a june blue

696
01:30:37.520 --> 01:30:44.800
 there the numerators are seven so let's look at the table check the denominator and i would hit

697
01:30:44.800 --> 01:30:52.400
 number 14 and 15 so it's certainly of the form s plus alpha all squared plus omega squared

698
01:30:52.400 --> 01:30:58.160
 doesn't matter that's negative just think of it as s plus minus three so it's s plus alpha all

699
01:30:58.160 --> 01:31:05.520
 squared plus omega squared so it's one of those two number 15 however though has the Laplace variable

700
01:31:05.520 --> 01:31:12.880
 has the s in the numerator our one doesn't it's just constant and that's what number 14 gives just

701
01:31:12.880 --> 01:31:20.640
 in the previous example so it's going to be number 14 but number 14 as i said with alpha equal to minus

702
01:31:20.640 --> 01:31:31.360
 three omega equal to four so that must mean that i need a four in the top line there to match the four

703
01:31:31.360 --> 01:31:38.400
 down here there's omega this is omega squared so omega's four so by number 14 i need an omega equal

704
01:31:38.400 --> 01:31:48.240
 to four up top there so i'll write that in and i'll multiply by my fraction to make sure that i'm not

705
01:31:48.320 --> 01:31:55.120
 changing things the fraction will have the thing that i want in the top line which is a seven and i'll

706
01:31:55.120 --> 01:32:02.160
 divide by what the table forced me to introduce which was a four because omega is equal to four

707
01:32:02.160 --> 01:32:07.840
 i had to write a four in the top line the table forced me to do that so the denominator here

708
01:32:07.840 --> 01:32:13.920
 must be a four the thing that i want on top which is seven down below the thing the table forced me

709
01:32:13.920 --> 01:32:21.600
 to introduce so i can then easily use this i've got seven over four times the inverse of this

710
01:32:21.600 --> 01:32:29.840
 this is just number 14 with alpha equal to minus three omega equal to four so i've got seven over

711
01:32:29.840 --> 01:32:38.560
 four e to the minus minus three t or e to the three t sine omega t omega's four so it's sine

712
01:32:38.560 --> 01:32:48.000
 40 so that is the answer there okay so that concludes this lecture so if we just have a quick

713
01:32:48.000 --> 01:32:53.920
 look at what we've done so we've introduced the inverse Laplace transform so you should now be

714
01:32:53.920 --> 01:33:01.040
 able to calculate the inverse Laplace transform of standard functions using the tables you should

715
01:33:01.040 --> 01:33:06.640
 be able to apply the first shifted theme to calculate the inverse Laplace transforms and you should

716
01:33:06.640 --> 01:33:13.120
 also be able to apply algebraic techniques which include partial fractions we looked at three different

717
01:33:13.120 --> 01:33:20.160
 types of partial fractions and also complete the square you'd be able to apply these methods to

718
01:33:20.160 --> 01:33:27.120
 calculate inverse Laplace transforms in the next class we're going to look at taking Laplace transforms

719
01:33:27.120 --> 01:33:35.600
 of derivatives and how this can be applied to solving basic ordinary differential equations

720
01:33:35.600 --> 01:33:43.520
 using Laplace transforms so just to have a quick look at what you should now be able to do

721
01:33:44.480 --> 01:33:51.120
 as far as the Mobius questions go you should be able to do questions one to nine and the tutorial

722
01:33:51.120 --> 01:33:57.200
 questions from the notes you should attempt question one parts one to eight and you should now be able

723
01:33:57.200 --> 01:34:04.960
 to do question two as well so that's the end of this lecture so we'll leave it at that for now

724
01:34:04.960 --> 01:34:09.520
 okay bye