1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Tmnsf orm Method 255

Solving for .C[y(x)], we find the Laplace transform of the solution, y(x), of the

given initial value problem satisfies

1

.C[y(x) ] = s 2 (s + 1) ·

Recalling the results from examples 1 and 2, we see that the so lution of the

given initial value problem is

y(x)=x-l+e-x.

EXERCISES 5.3

For each of the following functions H(s), use Laplace transform

information from Table 5.1 and the convolution theorem to find a

function h(x) such that .C[h(x)] = H(s). You may use a Computer

Algebra System to solve these exercises by defining the convolution

operator as in equation (1).

1 1

l.

s(s^2 + 9)

2.

(s+l)(s-2)

1

4.

1

3.

(s + l)(s - 2)^2 s(s^2 - 2s + 5)

s 1

5.

(s - l)(s^2 + 4)

6.

s^2 (s^2 - 4)

!Comments on Computer Software! In example 1, we calculated the con-
volution e - x x and found e x x = x - 1 + e-x. The following MAPLE
statement calculates and prints the convolution of x * e- x.

int( (x-xi) *exp(-xi) ,xi=O .. x);

As expected , the output is e - x + x - l.

In exercises 7-14 use the Laplace transform method and the con-

volution theorem to find the solution to the given initial value prob-

lem.

7. y' - 2y = 6;

8.

y(O) = 2

5

y(O) = 2

9. y" + 9y = 1; y(O) = 0, y'(O) = 0 (HINT: See exercise l.)


10. y" + 9y = l8e^3 x; y(O) = -1, y'(O) = 6