# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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k=1

• •

= (1/s) + ^^(-1)k e-ks/s = ?(-0 k]/s. We recognize k=1 k=0

the last infinite series as the geometric series, ^ark,

k=0

with a = 1 and r = -e s. This series converges to [1/(1+e-s) ] if |r| < 1. Hence,

?{f(t)} = (1/s)[1/(1+e-s)], s > 0.

Section 6.4

111

28. Using the definition of the Laplace transform we have

F(s) = ?{f(t)} = J0 e-stf(t)dt. Since f is periodic with period T, we have f(t+T) = f(t). This suggests that we

• -st

rewrite the improper integral as I e f(t)dt =

• (n+1)T -st

Ó InT e f(t)dt. The periodicity of f also suggests

n=0

that we make the change of variable t = r + nT. Hence,

• T • T

F(s) = Jo e-s(r+nT)f(r+nT)dr = (e sT) n J0 e-rsf(r)dr,

n=0 n=0

where we have used the fact that

f(r+nT) = f(r+(n-1)T) = ... = f(r+T) = f(r) from the definition that f is periodic. We recognize this last

series as the geometric series, aun, with

n=0

T

-rs -sT

a = |0e f(r)dr and u = e . The geometric series

converges to a/(1-u) for |u| < 1 and consequently we obtain

-sT -1 T -rs

F(s) = (1 - e-sT)-1 Jo e-rsf(r)dr, s > 0.

30. The function f is periodic with period 2. The result of

2 -st -2s

Problem 28 gives us ?{f(t)} = I e-stf(t)dt/(1-e- s).

Calculating the integral we have •2 1*2

J0 e stf(t)dt = J0 e stdt - J1 e stdt

2s -s

= (1-e ^/s + (e s-es)/s = (e-2s-2e-s+1)/s

-s 2

= (1-e ) /s. Since the denominator of ?{f(t)}, 1 - e-2s, may be written as (1-e-s)(1+e-s) we obtain the desired answer.

Section 6.4, Page 321

1. f(t) can be written in the form f(t) = 1 - up/2(t) and thus the Laplace transforms of the D.E. is (s2+1)Y(s) - sy(0) - y'(0) = (1/s) - e-Ps/2/s. Introducing

the I.C. and solving for Y(s), we obtain

Y(s) = (s2+1) -1 + [s(s2+1)] -1 - e-Ps/2/s(s2+1). Using partial fractions on the second and third terms we find

112

Section 6.4

Y(s) = (1/s) + (s2+1) -1 - s/(s2+1) - e-ps/2/s + e-ps/2s/(s2+1). The inverse transform of the first three terms can be obtained directly from Table 6.2.1. Using Theorem 6.3.1 to find the inverse transform of the last two terms we

have ?- 1{e-ps /2/s} = up/2(t)g(t - p/2) where g(t) = ?-1{1/s} = 1 and

?-1{e Ps/2s/(s2+1)} = up/2(t)h(t - p/2) where

h(t) = ?-1{s/(s2+1)} = cost. Hence, y = 1 + sint - cost + up/2(t)[cos(t - p/2) - 1]

= 1 + sint - cost - up/2(t)[1 - sint]. The graph of the forcing function is a unit pulse for 0 < t < p/2 and 0 thereafter. The graph of the solution will be composed of two segments. The first, for 0 < t < p/2, is a sinusoid oscillating about 1, which represents the system response to a unit forcing function and the given initial conditions. For t > p/2, the forcing function, f(t), is zero and the "initial" conditions are y(p/2) = lim 1 + sint - cost = 2 and

t^nr/2

y'(p/2) = lim cost + sint = 1. In this case the system

t^nr/2

response is y(t) = 2sint - cost, which is a sinusoid oscillating about zero.

3. According to Theorem 6.3.1,

?{u2p(t)sin(t-2p)} = e-2ps ?{sint} = e-2ps/(s2+1). Transforming the D.E., we have

(s2+4)Y(s) - sy(0) - y'(0) = 1/(s2+1) - e-2ps/(s2+1).

Introducing the I.C. and solving for Y(s), we obtain

-2 ps 2 2

Y(s) = (1-e s/(s +1)(s +4). We apply partial fractions to write

2 -1 2 -1 -2 ps 2 - 1 -2 ps 2 - 1

Y(s) = [s +1) - (s +4) - e (s +1) + e (s +4) ]/3.

We compute the inverse transform of the first two terms

directly from Table 6.2.1 after noting that

2-1 2 (s +4) = (1/2)[2/(s +4)]. We apply Theorem 6.3.1 to the

last two terms to obtain the solution,

y =(1/3){sint-(1/2)sin2t-u2p(t)[sin(t-2p)-(1/2)sin2(t-2p)]}.

This may be simplified, using trigonometric identities, to y = [(2sint - sin2t)(1-u2p(t))]/6. Note that the forcing function is sint - sin(t-2p) = 0 for t > 2p. The solution is y(t) = 2sint - sin2t for 0 < t < 2p. Thus

y(2p ) = 0 and y'(2p ) = 2cos2p - 2cos4p = 0. Hence the "initial" value problem for t > 2p is y" + 4y = 0, y(2p) = 0, y'(2p) = 0, which has the trivial solution y = 0.

Section 6.4

113

Taking the Laplace transform, applying the I.C. and using Theorem 6.3.1 we have (s2+s + 5/4)Y(s) = (1-e-ps/2)/s2. Thus

1-e-s/2

Y(s) =

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