1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Transform Method 263

cannot solve easily using the method of undetermined coefficients (See sec-

tion 4.4.). If we were to use the method of undetermined coefficients to solve

this initial value problem, we would have to consider the differential equation

of (2) to be three distinct differential equations defined on the three intervals

(-oo, 1], [1, 2], [2, oo)- see the definition of h(x) given in equation (1). Then

we would have to require that (i) the solution y 1 (x) on the interval (-oo, 1]

satisfy the given initial conditions, (ii) the solution y 2 (x) on the interval [1, 2]

satisfy y2(l) = y 1 (1) and y~(l) = y~(l), and (iii) the solution y 3 (x) on the

interval [2, oo] satisfy y3(2) = Y2(2) and y~(2) = y~(2). In this m anner, we

could obtain a solution,

!

Y1(x), -oo<x:::;l

y(x) = Y2(x), 1:::; x:::; 2

y3 (x), 2 :::; x < oo

of the given initial value problem which would be valid on ( -oo, oo). That is,

y(x) and y' (x) would be continuous on (-oo, oo )- in particular at x = 1 and

x = 2- and y(x) would satisfy the differential equation and initial conditions

of equation (2). Hence, in order to solve the initial value problem (2) using

the method of undetermined coefficients, we would in effect have to solve the

following three initial value problems in succession:

(3a) y~ + Y1 = O; Y1 (0) = 0, y~ (0) = 0, for - oo < x :::; 1

(3b) Y2 + Y2 = 2; Y2(l) = Y1(l), Y2(1) = y~(l), for 1:::; x:::; 2

(3c) y~ + y3 = 1; y3(2) = Y2(2), y~(2) = Y2(2), for 2:::; x < oo.

The advantage of the Laplace transform method is that the solution of the
initial value problem (2) can be obtained with one application of the method-
not three separate appli cations. In addition , it should be noted that the solu-
tion will also simultaneously satisfy equations (3a), (3b), and (3c). And, there-
fore, these equations can serve to check the validity of the Laplace transform
method solut ion. The following example illustrates how to use the Laplace
transform method to solve the initial value problem (2).

EXAMPLE 3 Solving an Initial Value Problem with a

Discontinuous Forcing Function

Use the Laplace transform method to solve the initial value problem

(2) y" + y = 2u(x - 1) - u(x - 2); y(O) = 0, y' (0) = 1.