1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans/ arm Method

EXAMPLE 2 Using the Laplace Transform to Solve a

Nonhomogeneous Linear Differential Equation

245

Find the general solution of the nonhomogeneous linear differential equation

y" + y' - 2y = x^2 - 1

using the Laplace transform method.

SOLUTION

Apply the Laplace Transform to the Differential Equation

Successively taking the Laplace transform of the given equation, using the
linearity property of the Laplace transform, replacing L'.[y"(x)] by the expres-
sion we obtain from equation (2) with n = 2, and replacing L'.[y'(x)] by the

expression we obtain from equation (2) with n = 1, results in

L'.[y" + y' - 2y] = L'.[x^2 - 1]

L'.[y"(x) ] + L'.[y'(x)] - 2.L'.[y(x)] = L'.[x^2 ] -L'.[1]

(5) (-y'(O) - sy(O) + s^2 L'.[y(x)]) + (-y(O) + sL'.[y(x)])-2.L'.[y(x)] =

2

s^3 - ~. s

Solve the Algebraic Equation for L'.[y(x)]

Letting A= y(O) and B = y'(O) and solving for L'.[y(x)], yields

2 1

A(s + 1) + B + 3 - -

(6) L'.[y(x)] = s s

s^2 + s - 2

A(s+l)+B 2

= (s + 2)(s - 1) + -s3-(s-+-2)-(s---l-)

1

s(s + 2)(s - 1)'

Expanding the right-hand side of this equation using partial fraction expan-
sion and combining like terms, we find

(7)

C1 C2 1 1 1

L'.[y(x)] = - + - - - - - - -

s + 2 s - 1 4s 2s2 s3

where c 1 = (4A - 4B - 1)/12 and c 2 = (2A + B + 1)/3 are arbitrary real

constants, since A and B are arbitrary real constants.