Page 69 - LAPLACE TRANSFORM

when s = 4 2s = A(s − 4) + B(s − 3)

2(4) = B(4 − 3)

B =

A B
X(S) = +
(s−3) (s − 4)
− 8
X(S) = +
(s−3) (s − 4)
4t
3t
x(t) = −6e + 8e

4.2 Second order differential equation

2
f’’(t) = s F(s) – sf(0) – f’(0) ; f’(t) = df(t) when t=0
dt

2
2
2
d x/dt = x’’(t) = s X(s) – sx(0) – x’(0) ; x’(t) = dx(t) when t=0
dt
2
2
2
d y/dt = y’’(t) = s Y(s) – sy(0) – y’(0) ; y’(t) = dy(t) when t=0
dt

Example 1

Find the differential equation x(t) for the following equation using Laplace transform.

x’’ – 4x’ + 3x = 0 ; given x (0) =5 and x’ (0) =7

Solution

x’’ – 4x’ + 3x = 0

2
{s X(s) – s x (0) – x’ (0)} – 4 {s X(s) – x (0)} + 3X(s) = 0
{s X(s) – s (5) – (7)} – 4 {s X(s) – (5)} + 3X(s) = 0
2

57