1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans! orm Method

h :=piecewise(x < 1, 0, 1 < = x and x < 2, 2, 2 < = x, 1):

h := convert(%, Heaviside);

DE:=diff(y(x), x$2) + y(x) = h;

laplace(DE, x, s);

L(y(x)) =solve(%, L(y(x)));

invlaplace(%, s, x);

267

The third and fourth statements specify the initial condition values. The fifth
statement defines t he piecewise function h( x), while the sixth statement tells
the computer to write h(x) in terms of Heaviside functions (unit step func-
t ions). MAPLE uses the notation "Heaviside(x - c)" instead of the notation
u(x - c) for the unit step function. The seventh statement defines the dif-
ferential equation to be solved. The eighth statement instructs the computer
to apply the Laplace transform to the differential equation and to substitute
the initial values into the transformed equation. The ninth statement tells
the computer to solve the output of the eighth statement algebraically for
L(y( x)). The last statement causes the computer to calculate the inverse
Laplace transform and to print the solution to the init ial value problem. The
solution printed is expressed in terms of Heaviside functions and is equivalent
to equation (7) of example 3.

5.5 Impulse Function

A force which is of relative large magnitude and which acts on a system for
a relative short period of time is call ed an impulse force. A golf club striking
a golf ball, a hammer striking a mass suspended on a spring, and a voltage
source connected to an electrical circuit for a short interval of time are all
examples of impulse forces. A function f(x) which represents an impulse force
is naturally called an impulse function. Since impulse functions represent
impulse forces, impulse functions are zero except for a short interval of time.
Suppose that f(x) is an impulse function and f(x) = 0 except for x 1 < x < x 2.
Then the integral

I1 = 1-: f(x) dx = 1~

2

f(x) dx

is called the total impulse of the function f(x) and represents the total force
imparted to the system.

Let us consider the set of impulse step functions d<(x) defined by

{

1 /c:,

d<(x) =

0,

O<x<c:

x:::; 0 or x 2 E