# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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156

Chapter 3. Second Order Linear Equations

Find the general solution of

y" + y + y = 0. (18)

The characteristic equation is

r2 + r + 1 = 0,

and its roots are

_ 1 ± (1 4)1/2 1 ,

r = 2 =~2 1 ~ .

Thus k = 1/2 and x = V3/2, so the general solution of Eq. (18) is

y = cxe~t/2 cos(V3t/2) + c2e~t/2 sin(V3t/2). (19)

EXAMPLE

2

Find the general solution of

y" + 9y = 0. (20)

The characteristic equation is r2 + 9 = 0 with the roots r = ±3i ; thus X = 0 and ii = 3. The general solution is

y = Cjcos3i + c2sin3i ; (21)

note that if the real part of the roots is zero, as in this example, then there is no exponential factor in the solution.

Find the solution of the initial value problem

16/'- 8/ + 145y = 0, y(0) = -2, /(0) = 1. (22)

The characteristic equation is 16r2 8r + 145 = 0 and its roots are r = 1/4 ± 3i. Thus the general solution of the differential equation is

y = clet/4 cos 3t + c2et/4 sin 3t.

(23)

To apply the first initial condition we set t = 0 in Eq. (23); this gives

y(0) = c1 = -2.

For the second initial condition we must differentiate Eq. (23) and then set t = 0. In this way we find that

y'(0) = 1 c1 + 3c2 = 1, from which c2 = 1/2. Using these values of c1 and c2 in Eq. (23), we obtain

y = -2et/4cos3t + 2 et/4sin3t

(24)

as the solution of the initial value problem (22).

We will discuss the geometrical properties of solutions such as these more fully in Section 3.8, so we will be very brief here. Each of the solutions u and v in Eqs. (15) represents an oscillation, because of the trigonometric factors, and also either grows or

3.4 Complex Roots of the Characteristic Equation

157

decays exponentially, depending on the sign of X (unless X = 0). In Example 1 we have X = 1/2 < 0, so solutions are decaying oscillations. The graph of a typical solution of Eq. (18) is shown in Figure 3.4.1. On the other hand, X = 1/4 > 0 in Example 3, so solutions of the differential equation (22) are growing oscillations. The graph of the solution (24) of the given initial value problem is shown in Figure 3.4.2. The intermediate case is illustrated in Example 2 in which X = 0. In this case the solution neither grows nor decays exponentially, but oscillates steadily; a typical solution of Eq. (20) is shown in Figure 3.4.3.

FIGURE 3.4.2 Solution of 16/' 8/ + 145y = 0, y (0) = -2, y'(0) = 1

158

Chapter 3. Second Order Linear Equations

PROBLEMS In each of Problems 1 through 6 use Eulers formula to write the given expression in the form a + ib.

1. exp(1 + 2i) 2. exp(2 3i)

5. 21

-l+2i

In each of Problems 7 through 16 find the general solution of the given differential equation.

7. y' - 2y + 2 y 0 8. y " 2y' + 6y 0

9. y' + 2y o II 00 1 10. y " + 2y' + 2 y 0

11. y' + 6y + 13y 0 12. 4y ' + 9y 0

13. y' + 2y + 1.25 y 0 14. 9y O II ? 1 +

15. y' + y' + 1.25y 0 16. y " + 4y' + 6.25y 0

In each of Problems 17 through 22 find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior for increasing t.

17

18

19

20 21 22

23

? 24.

y" + 4y = 0, y (0) = 0, y'(0) = 1

y" + 4y' + 5 y = 0, y (0) = 1, y'(0) = 0

y" 2y; + 5y = 0, y(n/2) = 0, y (n/2) = 2

y" + y = 0, y(n/3) = 2, y'(n/3) = 4

y"+ y'+ 1.25y = 0, y(0) = 3, y'(0) = 1

y" + 2y; + 2y = 0, y(n /4) = 2, y' (n /4) = 2

Consider the initial value problem

3un u/ + 2u = 0,

m(0) = 2, u (0) = 0.

(a) Find the solution u (t) of this problem.

(b) Find the first time at which |u (t) | = 10. Consider the initial value problem

5u" + 2u + 7u = 0, u(0) = 2, u (0) = 1.

(a) Find the solution u (t) of this problem.

(b) Find the smallest T suchthat lu(t)l <0.1forall t > T.

? 25. Consider the initial value problem

y" + 2y! + 6y = 0, y(0) = 2, y'(0) = a > 0.

3 en 4 e2-(n/2)t

(a) Find the solution y (t) of this problem.

(b) Find a so that y = 0 when t = 1.

3.4 Complex Roots of the Characteristic Equation

159

(c) Find, as a function of a, the smallest positive value of t for which y = 0.

(d) Determine the limit of the expression found in part (c) as a ^ X.

? 26. Consider the initial value problem

y" + lay' + (a2 + 1)y = 0, y(0) = 1, y'(0) = 0.

(a) Find the solution y(t) of this problem.

(b) For a = 1 find the smallest T such that ly(t)l < 0.1 for t > T.

(c) Repeat part (b) for a = 1 /4, 1/2, and 2.

(d) Using the results of parts (b) and (c), plot T versus a and describe the relation between T and a.

27. Show that W (ekt cos xt, ekt sin xt) = xe2kt.

28. In this problem we outline a different derivation of Eulers formula.

(a) Show that y: (t) = cos t and y2(t) = sint are a fundamental set of solutions of y" + y = 0; that is, show that they are solutions and that their Wronskian is not zero.

(b) Show (formally) that y = e11 is also a solution of y" + y = 0. Therefore,

eU = Cj cos t + c2 sin t (i)

for some constants Cj and c2. Why is this so?

(c) Set t = 0 in Eq. (i) to show that c1 = 1.

(d) Assuming that Eq. (14) is true, differentiate Eq. (i) and then set t = 0 to conclude that c2 = i. Use the values of c1 and c2 in Eq. (i) to arrive at Eulers formula.

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