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

ISBN 0-471-31999-6

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39. By direct substitution, or from Problem 38, x = lnt will

2 2

transform the D.E. into d y/dx + y = 0, since a = 1 and b = 1. Thus y = c1cosx + c2sinx, with x = lnt, t > 0.

Section 3.5

Section 3.5, Page 166

14.

rt

Substituting y = e

2

r - 2r + 1 = 0, which gives r1

into the D.E., we find that

1 and r

2

1. Since the

roots are equal, the second linearly independent solution is tet and thus the general solution is y = c1et + c2tet.

The characteristic equation is 25r - 20r + 4 = 0, which

2

may be written as (5r-2) r1,r2 = 2/5. Thus y = c1e

0 and hence the roots are

2t/5

+ c2 te

2t/5

12. The characteristic equation is r2 - 6r + 9 = (r-3)2, which has

the repeated root r = 3. Thus

3t 3t , . ,

y = c1e + c2te , which gives

y(0)

0, y'(t)

, 3t 04_ 3t x

c2 (e +3te )

and y'(0) = c2

3t

2. Hence

y(t)

2te^

The characteristic equation is

22 r + 4r + 4 = (r+2) = 0, which

has the repeated root r = -2.

Since the I.C. are given at

t = -1, write the general solution

Then

as y

-2(t + 1) -2(t + 1)

? + c2te

0 -2(t+1) -2(t+1) ~ . -2(t + 1) , ^

y = -2c1e + c2e - 2c2te and hence

2 and -2c1+3c2

1 which yield c1 = 7 and c2

5.

Thus y = 7e 2(t+1) + 5te 2(t+1), a decaying exponential as shown in the graph.

17a. The characteristic equation is 4r + 4r + 1 = (2r+1) = 0,

-t/2

so we have y(t) = (c1 + c2t)e . Thus y(0) = c2 = 1 and y'(0) = -cx/2 + c2 = 2 and hence c2 = 5/2 and

y(t)

(1 + 5t/2)e

t/2

17b. From part (a), y'(t) =

1 -t/2 5 -t/2

— (1 + 5t/2)e ' + —e ' = 0, when

22

1 5t 5 8 -4/5

- — - --------- + — = 0, or t0 = — and y0 = 5e .

2 4 2 0 5 0

c

1

c1 c2

48

Section 3 . 5

11 17c. From part (a), - — + c2 = b or c2 = b + — and

2 2 2 2

y(t) = [1 + (b + 1)t]e-t/2.

2

. 1 1 -t/2 1-t/2

17d. From part (c), y(t) = - — [1 + (b+— )t]e ' + (b+— )e ' = 0

2 2 2

4b

which yields tM = ---------- and

11 2b+1

,, 2b + 1 4b -2b/(2b + 1) -2b/(2b+1)

yM = (1 + -------- • )e ' = (1 + 2b)e .

1 2 2b+1

r t

19. If r1 = r2 then y(t) = (c1 + c2t)e 1 . Since the exponential

is never zero, y(t) can be zero only if c1 + c2t = 0, which

yields at most one positive value of t if c1 and c2 differ

in sign. If r2 > r1 then

r,t r2t , (r,-r,)t. , .

y(t) = c1e 1 + c2e 2 = e 1 (c1 + c2e 2 1 ). Again, this is

zero only if c1 and c2 differ in sign, in which case

ln(-c1/c2)

t =

(r2-r1)

21. If r2 n r1 then f(t;r1,r2) = (er2t - er1t)/(r2 - r1) is

defined for all t. Note that f is a liner combination of

r t r t

two solutions, e 1 and e 2 , of the D.E. Hence, f is a solution of the differential equation. Think of r1 as fixed and let r2 —— r1. The limit of f as r2 —— r1 is

indeterminate. If we use L'Hopital's rule, we find

r2t r,t . r2t

e 2 - e 1 te 2 r t

lim ---------------- • = lim --------- = te 1 . Hence, the

r2 — r1 r2 - r1 r2 — r1 1

rt

solution f(t;r1,r2) — te 1 as r2 — r1.

25. Let y2 = v/t. Then y'2 = v'/t - v/t2 and

y2 = v"/t - 2v'/t2 + 2v/t3. Substituting in the D.E. we obtain

t2 (v"/t - 2v'/t2 + 2v/t3) + 3t(v'/t - v/t2) + v/t = 0. Simplifying the left side we get tv" + v' = 0, which yields v' = c1/t. Thus v = c1lnt + c2. Hence a second solution is y2(t) = (c1lnt + c2)/t. However, we may set c2 = 0 and c1 = 1 without loss of generality and thus we have y2(t) = (lnt)/t as a second solution. Note that in

Section 3.5

49

the form we actually calculated, y2(t) is a linear combination of 1/t and lnt/t, and hence is the general solution.

27. In this case the calculations are somewhat easier if we

2

do not use the explicit form for y1(x) = sinx at the beginning but simply set y2(x) = y1v. Substituting this form for y2 in the D.E. gives x(y1v) "-(y1v)/+4x3(y1v) = 0.

On carrying out the differentiations and making use of the fact that y1 is a solution, we obtain

// f /

xy1v + (2xy1 - y1)v = 0. This is a first order linear

equation for v', which has the solution v' = cx/(sinx2) 2.

2

Setting u = x allows integration of this to get 2

v = c1 cotx + c2. Setting c1 = 1, c2 = 0 and multiplying

by y1 = sinx2 we obtain y2(x) = cosx2 as the second solution of the D.E.

30. Substituting y2(x) = y1(x)v(x) in the D.E. gives

2 n f 2 1

x (y-iv)" + x(y-,v)' + (x - —) y-,v = 0. On carrying out the 1 1 4 1

differentiations and making use of the fact that y1 is a solution, we obtain x2y1v" + (2x2y1 + xy1)v' = 0. This is a first order linear equation for v ,

v" + (2y1/y1 + 1/x)v' = 0, with solution

/

y1 1

v (x) = cexp[-J (2— + — )dx] = cexp[-2lny1 - lnx]

y1 x 1

1c 2

= c = ------------------ = c csc x,

2 ,-1.2. xy1 x(x sin x)

where c is an arbitrary constant, which we will take to be

one. Then v(x) = Jcsc x dx = -cotx + k where again k is an

arbitrary constant which can be taken equal to zero. Thus

-1/2 -1/2 y2(x) = y1(x)v(x) = (x sinx)(-cotx) = -x cosx. The

-1/2

second solution is usually taken to be x cosx. Note that c = -1 would have given this solution.

31b. Let y2(x) = exv(x), then y2 = exv' + exv, and

y2 = exv" + 2exv' + exv. Substituting in the D.E. we

obtain xexv" + (xex-Nex)v' = 0, or v" + (1-N/x)v' = 0.

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