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General Procedure

is called the transformed equation.

2. Obtain an equation L(y) = F(s), where F(s) is an algebraic expression in
the variable s.
−1
3. Apply the inverse transform to yield the solution y = L (F(s)).
The various techniques for determining the inverse transform include partial

fraction decomposition, translation, derivative and integral theorems,
convolutions, and integration in the complex plane. All of these techniques
except the latter are used in conjunction with standard tables of Laplace
.
transforms.

Example 5.2, Solve

t
′′′
′
′′
′′
y + y = e + t + 1, y(0) = y (0) = y (0) = 0.
. . . . .
.
Solution. Taking L of both sides gives
t
′′
′′′
L(y ) + L(y ) = L(e ) + L(t) + L(1),
or

1 1 1
2
2
3
′
′′
′
[s L(y)−s y(0)−sy (0)−y (0)]+[s L(y)−sy(0)−y (0)] = + + .
s − 1 s 2 s
Putting in the initial conditions gives
2
2s − 1
2
3
s L(y) + s L(y) ,
2
s (s − 1)
which is
2
2s − 1
L(y) = .
4
s (s + 1)(s − 1)
Applying a partial fraction decomposition to
1
2s − 1 A B C D E F
L(y) = = + + + + + ,
4
s (s + 1)(s − 1) s s 2 s 3 s 4 s + 1 s − 1
we find that
1 1 1 1
L(y) = − + − + ,
s 2 s 4 2(s + 1) 2(s − 1)

and consequently

( ) ( ) ( ) ( )
1 1 1 1 1 1
−1 −1 −1 −1
y = −L + L − L + L =
s 2 s 4 2 s + 1 2 s − 1
1 1 1
3
t
= −t + t − e −t + e .
6 2 2
. . . .

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