1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Transform Method 243

du= -se-sxdx and v = y'(x). Substituting these expressions into the formula

for integration by parts, yields

If y'(x) is of exponential order a as x , +oo, then for s >a, y'(x)e-sx , 0

as x ____, +oo and, therefore,

.C[y"(x) ] = -y'(O) - sy(O) + s^2 .C[y(x)] for s >a.

By induction, we obtain the following general formula for the Laplace trans-

form of then-th derivative of the function y(x).

Laplace Transform of the n-th Derivative of y(x), .C[y(nl(x)]

Let y(x), y(ll(x), ... , y(n-ll(x) be continuous on [O, oo) and let y(nl(x) be

piecewise continuous on [O, oo). FUrthermore, let y(x), y(^1 l(x), ... , y(nl(x)

be of exponential order a. Then, for s > a

Since the solutions of homogeneous linear differential equations with con-
stant coefficients and all of their derivatives are continuous and of exponential
order a as x , +oo for some constant a, equation (2) is valid for the all n-th
order linear homogeneous differential equations with constant coefficients. If,
in addition, the nonhomogeneity b(x) of a nonhomogeneous linear differential
equation with constant coefficients is of exponential order a as x , +oo, then
b(x) has a Laplace transform and equation (2) is valid for that nonhomoge-
neous differential equation.
The following example illustrates how to use the Laplace transform method
to obtain the general solution of a second-order linear homogeneous differential
equations with constant coefficients.

EXAMPLE 1 Using the Laplace Transform to Solve a

Homogeneous Linear Differential Equation

Use the Laplace transform method to solve the homogeneous linear differ-

ential equation

y" + 4y = 0.