Page 77 - LAPLACE TRANSFORM

Example 1

Solve the differential equation dy + y = 3 by using the Inverse Laplace
dt
Transform. Given that y(0)= 0 when t=0.

STEP 1: Take the Laplace Transform for both sides.

L [ ] )(ty ' + y = L [ ] 3

STEP 2: Substitude the theorem for Left Hand Side (LHS) .

Find the Laplace for Right Hand Side (RHS)
3
0
[sY ( s) − y ) + ] y =
(
s

STEP 3 : Substitude the values of y(0)=1, t=0
3
[sY( s) − 0 + ] y =
s
STEP 4 : Solve for Y(s)
3
[sY (s ) + y ] =
s
3
Y (s )[ + ] 1 =
s
s
3
Y (s ) =
s
s ( + ) 1
STEP 5 : Solve for the inverse , ( ) = − [ ( )] using Partial Fraction.
 3 
y ) (t = L −1  
 s (s + )1 
3 = A + B
s (s + ) 1 s (s + ) 1

3 = A (s + ) 1 + B (s ) ………………equation 1
If s= 0

3 = A 0 ( + ) 1 + ) 0 ( B
3 = ) 1 ( A

A = 3
If s= -1
3 = A (− 1+ ) 1 + B (− ) 1

3 = B (− ) 1
B = − 3

70