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

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f (t + n) = f (t).

See Figure 6.3.9.

2n 3n t

FIGURE 6.3.9 A rectified sine wave.

33. (a) If f (t) = 1 — u1(t), find L{ f (t)}; compare with Problem 24. Sketch the graph of Ó = f (t). t

(b) Let g(t) = f (f) df, where the function f is defined in part (a). Sketch the graph

*^0

of y = g(t) and find L{g(t)}.

(c) Let h(t) = g(t) — u1 (t)g(t — 1), where g is defined in part (b). Sketch the graph of y = h(t) and find L{h(t)}.

34. Consider the function p defined by

p(t) =

t,

2 t,

0 < t < 1, 1 < t < 2;

P(t + 2) = p(t).

(a) Sketch the graph of y = p(t).

(b) Find L{p(t)} by noting that p is the periodic extension of the function h in Problem 33(c) and then using the result of Problem 28.

6.4 Differential Equations with Discontinuous Forcing Functions

317

(c) Find C{p(t)} by noting that

p(t) = f f(t) dt,

Jo

where f is the function in Problem 30, and then using Theorem 6.2.1.

6.4 Differential Equations with Discontinuous Forcing Functions

In this section we turn our attention to some examples in which the nonhomogeneous term, or forcing function, is discontinuous.

1

Find the solution of the differential equation

2/' + / + 2 y = g(t), (1)

where

g(t) = è^ - u2°(t) = { 0, 0 <?< 5’ and t > 20. (2)

Assume that the initial conditions are

y(0) = 0, Ó (0) = 0. (3)

This problem governs the charge on the capacitor in a simple electric circuit with a unit voltage pulse for 5 < t < 20. Alternatively, y may represent the response of a damped oscillator subject to the applied force g(t).

The Laplace transform of Eq. (1) is

2s2 Y(s) - 2sy(0) - 2Ó(0) + sY(s) - y(0) + 2Y(s) = ÝÄÎ} - Öè^)}

= (e~5s - e-20s)/s.

Introducing the initial values (3) and solving for Y(s), we obtain

e-5s _ e-20s

Y(s) = e 2 e-------------------------------------------------. (4)

s (2s + s + 2)

To find y = ô^) it is convenient to write Y(s) as

Y(s) = (e-5 - e-20s) H(s), (5)

where

H (s) = 1/s (2s2 + s + 2). (6)

Then, if h(t) = C-1{H(s)}, we have

y = ô(0 = è5(t)h(t - 5) - u20(t)h(t - 20). (7)

318

Chapter 6. The Laplace Transform

Observe that we have used Theorem 6.3.1 to write the inverse transforms of e—5s H(s) and e~20sH(s), respectively. Finally, to determine h(t) we use the partial fraction expansion of H(s):

a bs + c

H (s) = —+ TT2---------------Ã • (8)

s 2s + s + 2

Upon determining the coefficients we find that a = 2, b =— 1, and c =— 1. Thus

1 \2 ³ 15 ’ (9)

s

so that, by referring to lines 9 and 10 of Table 6.2.1, we obtain

h(t) = 1 — 2[e—tj4 cos(VT5 tj4) + (VT5 j15)e—tj4 sin^VT51 j4)]. (10)

In Figure 6.4.1 the graph of y = ô(³) from Eqs. (7) and (10) shows that the solution consists of three distinct parts. For 0 < t < 5 the differential equation is

2 Ó + / + 2 y = 0 (11)

and the initial conditions are given by Eq. (3). Since the initial conditions impart no energy to the system, and since there is no external forcing, the system remains at rest; that is, y = 0 for 0 < t < 5. This can be confirmed by solving Eq. (11) subject to the initial conditions (3). In particular, evaluating the solution and its derivative at t = 5, or more precisely, as t approaches 5 from below, we have

y(5) = 0, y (5) = 0. (12)

Once t > 5, the differential equation becomes

2/ + / + 2 y = 1, (13)

whose solution is the sum of a constant (the response to the constant forcing function) and a damped oscillation (the solution of the corresponding homogeneous equation). The plot in Figure 6.4.1 shows this behavior clearly for the interval 5 < t < 20. An expression for this portion of the solution can be found by solving the differential equation (13) subject to the initial conditions (12). Finally, for t > 20 the differential equation becomes Eq. (11) again, and the initial conditions are obtained by evaluating the solution of Eqs. (13), (12) and its derivative at t = 20. These values are

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