# Teacher's Guide to Laplace Transform

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```					Explanatory Notes to Laplace Transform

2004/2005

Laplace Transform Page 4 E.g. 2 Use integration by parts Page 8 E.g. 5 i) ii) iii)
2 3 2 5    s s 2 s3 s  4 1 3 2   3 2 2 s (s  4) s 4 2 2(s  4) 12s   4 2 2 s ( s  4)  9 (s  4)2

Inverse Laplace Transform Page 14 E.g. 6 L1( Page 16 E.g. 7 L1(
3s  1
2

1 2 )  L1(  )  e t  2et s 1 s 1 s 1

4 5 4 )  L1(   )  4  5t  4et 2 2 s 1 s s s (s  1)

s5

Page 17 E.g. 8 4s  5 5  5s  16 5 5 s 1 5 5 L1( )  L1(  )   L1( )  4L1( )   cos2t  2 sin 2t 4s 4(s 2  4) 4 4 4 4 s 3  4s s2  4 s2  4

P. 1

Explanatory Notes to Laplace Transform

2004/2005

Solve Differential Equations Page 20 E.g. 1 a) sF (s)  2 Page 23 E.g. 3 Phase 1
s 2 X ( s )  sx (0)  x' (0)  4[ sX ( s )  x(0)]  5 X ( s )  0 ( s 2  4 s  5) X ( s )  1  0 Phase 2 1 1 X ( s)   s 2  4s  5 (s  2)2  1 Phase 3

b) s 2 F (s)  2s  3

c) (3s 2  2s  1) F (s)  6s  13

Take Inverse Laplace, x(t )  e2t sin t

Practical Example 1 ODE : 15i  0.02 Phase 1 Phase 2 Phase 3
di  24 with i(0)=0 dt

15I ( s)  0.02[ sI ( s)  i(0)] 

24 s 24 1 1200 1 1 I ( s)   (  ) s 0.02s  15 750 s s  750 1200 i (t )  (1  e750t ) 750

Practical example 2
1 idt  24 with i(0)=0 0.1 I ( s) 24 Phase 1 5I ( s)  10  s s 24 1 24 1 Phase 2 I ( s)   s 5  10 / s 5 s2 24  2t e Phase 3 i (t )  5

ODE : 5i 



P. 2

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Lingjuan Ma