1550078481-Ordinary_Differential_Equations__Roberts_

266 Ordinary Differential Equations

3. Use Theorem 5.3 and Table 5.1 to find an inverse Laplace transform of
   the following functions.

e-s

b.

1 - e-2s

a.

s+2^32

se-7rs

d.

se-7rs

c.

s^2 + 9 s^2 -^9

e-2s

f.

se-3s

e.

s^2 + 2s + 2 s2 + 2s +^2

e-3s

h.

e-s

g.

s^2 + 2s - 3 s^2 - 2s + 1

4. Solve the following initial value problems. The functions fi(x) a re as
   defined in Exercise 1.

a. y' + 2y = fi(x); y(O) = 1

b. y" - y' - 2y = f2(x); y(O) = 0, y'(O) = 1

c. y" - 2y' = f3(x); y(O) = 1, y'(O) = 0

d. y" - 2y' + y = f4(x); y(O) = 0, y'(O) = 1

e. y^11 +4y=f5(x); y(O)=l, y'(O)=l

g. y"-4y=f6(x); y(O)=O, y'(O)=O

g. y" - 4y' + 5y = h(x); y(O) = 1, y'(O) = 0

IConnents on Computer Software! In example 3, we solved the initial
value problem y" + y = h(x); y(O) = 0, y'(O) = 1, where h( x) is the
piecewise defined function

l

0, x < 1

h(x) = 2, 1 ::::; x < 2

1, 2::::; x

The following ten MAPLE statements a lso solve this initial value problem and
output some important intermediate results.

with(inttrans):

a li as(L(y(x)) =laplace(y(x), x, s)):

y(O) := 0:

D(y)(O) := 1: