Page 821 - Advanced_Engineering_Mathematics o'neil

Answers to

Selected Problems

CHAPTER ONE FIRST-ORDER DIFFERENTIAL EQUATIONS
Section 1.1 Terminology and Separable Equations
1. Yes, since
√ 1

2ϕ(x)ϕ (x) = 2 x − 1 √ = 1for x > 1.
2 x − 1
2
3
2
3. Yes 5. Yes 7. y = 2x + c,or y = (2x + c) 1/3
9. Not separable
11. y = 1/(1 − cx); also y = 0and y = 1 as singular solutions
13. sec(y) = kx and y = (2n + 1)π/2 with n any integer
2
15. Not separable 17. 1 y − y + ln(y + 1) = ln(x) − 2
2
2
2
2
19. (ln(y)) = 3x − 3 21. 3y sin(3y) + cos(3y) = 9x − 5
23. 45 F 25. 10(1/2) 1/4.5 ≈ 8.57 kg
◦
√
29. The tank empties at t = 3888 2 seconds; about 91 minutes and 39 seconds.
31. (a) t = 243(64/27) = 576 seconds, or 9 minutes, 36 seconds (b) t = (160/27)(1296/5) = 1536 seconds; about 25
minutes, 36 seconds.
Section 1.2 Linear Equations
3 3 1 1 −2x 2 2x
1. y = cx + 2x ln|x| 3. y = x − + ce 5. y = 4x + 4x + 2 + ce
2 4
2
2
7. y = x − x − 2 9. y = x + 1 + 4(x + 1) −2 11. y =−2x + cx
13.
A 1 (t) = 50 − 30e −t/20 , A 2 (t) = 75 + 90e −t/20 − 75e −t/30 ,
A 1 (t) has its minimum value of 5450/81 pounds at 60ln(9/5) minutes.
Section 1.3 Exact Equations
2
xy
2
1. 2xy + e + y = c 3. not exact
2
3
2
3
5. y + xy + ln|x|= c 7. α =−3; x y − 3xy − 3y = c
4
9. 3xy − x = 47 11. x sin(2y − x) = π/24
13. The general solutions are the same, since the equations ϕ(x, y) = k and ϕ(x, y) + c = k implicitly define the same
functions of y in terms of x.
15. μ(x, y) = x 1/2 y 3/2 ,(1/5)x 5/2 y 5/2 + (1/3)x 3/2 = c
Section 1.4 Homogeneous, Bernoulli and Riccati Equations
1. Riccati equation with S(x) = x;
x
y = x +
c − ln(x)
801

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October 14, 2010 17:50 THM/NEIL Page-801 27410_25_Ans_p801-866