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

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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.

y

2 -

1 1 4 6 1

\ 1 1

2 8 ¥

FIGURE 3.4.1 A typical solution of y" + y' + y = 0.

FIGURE 3.4.2 Solution of i6y" - 8y' + i45y = 0, y (0) = -2, y'(0) = ³

158

Chapter 3. Second Order Linear Equations

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

1. exp(1 + 2i) 2. exp(2 — 3i)

5. 21

-1+2i

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

7. yII --- 2y' + 2 y --- 0 8. y" --- 2 y I + 6 y = 0

9. yII + 2 y I --- 8 y = 0 10. y" + 2 y I + 2 y = 0

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

13. yII + 2 y I + 14. 9y +

2 Iy9

y ---

II y4

0 =

0

15. yII + y I + 1 . 25 y = 0 16. y " + 4 yI + 6. 25 y = 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,

u(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 such that lu(t)l <0.1 forall t > T.

> 25. Consider the initial value problem

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

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

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

3.4 Complex Roots ofthe 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 ^ to.

> 26. Consider the initial value problem

y" + 2ay' + (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 (eu cos ät, eu sin ät) = äå2è.

28. In this problem we outline a different derivation of Euler’s formula.

(a) Show that y1 (t) = cos t and y2(t) = sin t 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,

en = c1 cos t + c2 sin t (i)

for some constants c1 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 Euler’s formula.

29. Using Euler’s formula, show that

cos t = (eit + e^it )/2, sin t = (eit — e—it)/2i.

30. If ert is given by Eq. (13), show that e(r1+r2)t = er1ter2t for any complex numbers r1 and r2.

31. If ert is given by Eq. (13), show that

— ert = rert dt

for any complex number r .

32. Let the real-valued functions p and q be continuous on the open interval I, and let y = ô() = u(t) + i v(t) be a complex-valued solution of

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