Page 83 - LAPLACE TRANSFORM

Solve the differential equation − 3 − 10 = 2 (0) = 1, (0) = 2
′
′
′′

STEP 1: Take the Laplace Transform for both sides.
1
−
=
if
s
L [ ] [ ] 10)(3)(ty ' ' − L y ' t − [ ] [ ] 2)(ty = L

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

Find the Laplace for Right Hand Side (RHS)
] 2
−
[ Ys 2 s ) ( − sy ) 0 ( − ) 0 ( ' y ] [ (3 sY s ) − ) 0 ( y ] 10 [ (sY ) =
−

STEP 3 : Substitude the values of y(0)=1, y (0)=2
’
[ Ys 2 s ) ( − s ) 1 ( − 2 − s ) − 1 − [ (sY ) =
] 10
] [ (3 sY
] 2

STEP 4 : Solve for Y(s)
s 2 Y (s ) − s − 2 − 3sY (s ) + 3 − 10Y (s ) = 2
s
2
Y (s )[s 2 − 3 −s 10 ] = + s + 2 − 3
s
s 2 − s + 2
Y (s ) =
s ( −s 5 )( +s ) 2
STEP 5 : Solve for the inverse , ( ) = − [ ( )] using Partial Fraction.

s 2 − s + 2 A B C
= + +
s ( −s 5 )( +s ) 2 s ( −s ) 5 ( +s ) 2
s 2 − + s 2 = A (s )( 5 − + s ) 2 + B (s )( +s ) 2 +c (s )(s ) 5 − …………….equation (1)

If s=-1 if s=-5

) 0 ( 2 − 0 + 2 = A 0 ( − 5 )( 0 + ) 2
) 5 (
2 = A (− 5 )( ) 2 2 − 5+ 2 = 5 ( B )( 5+ ) 2

2 = A (− 10 ) 22 = 5 ( B )( ) 7
2 1 22
A = − = − B =
10 5 35
If s = -2

2
(− ) 2 − (− ) 2 + 2 = C (− 2 )(− 2 − ) 5
8 = C (− 2 )(− ) 7
8 4
C = =
14 7

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