1550078481-Ordinary_Differential_Equations__Roberts_

240 Ordinary Differential Equations

5. Find .C[h(x)] for

l

x O-S:x< 2

h(x) = - x ~ 4, 2-:;: x < 4

0, 4-:::: x

6. Find .C[k(x)] for

l

0, 0-:::: x < 1

k(x)= 1, 1-S:x<2

0, 2-:::: x

7. Use Table 5.1 and the linearity property of Laplace transforms to find
   the following transforms.

a. .C[5] b. .C[e]

c. .C[3x - 2] d. .C[e-x(2x + 1)]

e. .C[e^2 x sin 3x - 2 cos x]

I Comments on Computer Software I The following two MAPLE state-

ments may be used to calculate the Laplace transform of f(x) = x^2 sin x.

with(inttrans):

F :=laplace(x /\ 2 * sin(x), x, s );

The output displayed by MAPLE is

-1+3s^2

F := 2 (1 + s2)3

The first statement above, with(inttrans):, instructs the computer to

load software for calculating the Laplace transform.

The following four statements calculate the Laplace transform of the

piecewise continuous function of exercise 4.

with(inttrans):

g:=piecewise(O <= x and x < 1, 1 - x, 1 <= x, x - 1):

H: =convert(g, Heaviside):

G:=laplace(H, x, s);