Page 822 - Advanced_Engineering_Mathematics o'neil

802 Answers to Selected Problems

3. Bernoulli equation with α = 2;
1
y =
1 + ce x 2 /2
5. Homogeneous; y ln|y|− x = cy
2
2
7. Exact with general solution implicitly defined by xy − x − y = c; also homogeneous
9. Bernoulli with α =−3/4; 5x 7/4 y 7/4 + 7x −5/4 = c
11. Bernoulli with α = 2;
2
y = 2 +
2
cx − 1
x
13. Riccati with S(x) = e ;
2e x
y =
2x
ce − 1
17. Choose h = 2,k =−3 to obtain
2
2
3(x − 2) − 2(x − 2)(y + 3) − (y + 3) = k
2
19. (2x + y − 3) = k(y − x + 3)
Section 1.5 Additional Applications
1. Maximum height is 342.25 feet, at t = 1/8 second; at t = 19/4 the bag hits the ground with speed 148 ft/sec.
−t
3. Terminal velocity is 8 ft/sec. Distance fallen is 32(t − 1 + e ) for 0 ≤ t ≤ 4, and for t ≥ 4,
2

−4
32(3 + e ) + 8(t − 4) + 2ln(1 − e −8(t−4) ) − 2ln ,
5 − 4e −4
−4
−4
where k = (3 − 4e )/(5 − 4e ).
5. Surfacing velocity is approximately 17.5 ft/sec at t ≈ 10.56 sec.
√
7. t = 2 R/g, with R the radius of the circle.
−5
9. Voltage reaches 76 volts when t = (1/2)ln(20). Current at this time is 32(10 )e −ln(20) = 16 micro amps.
11. The inductive time constant is
L e(E − Ri(0))
ln ,
R E
decreasing as i(0) is chosen larger.
13. y =−(3/4)ln|x|+ c
2
2
15. (y − 1) =−(1/2)x + c, a family of ellipses
2
2
17. y (ln(y ) − 1) = c − 2x 2
√
−θ
19. (a) Pursuit curves are r = f (θ) = (a/ 2)e . (b) distance = a (c) No
√ √ √
21. Time is ( 3/2)(ln(6 + 35), about 2.15 seconds; velocity is 12 5, about 26.84 ft/sec.
Section 1.6 Existence and Uniqueness Questions
1. f (x, y) = sin(xy) and ∂ f/∂y = x cos(xy) are continuous for all (x, y), hence in any rectangle centered at (π/2,1).
2
2
2
3. f (x, y) = x − y + 8x/y and ∂ f/∂y =−2y − 8x/y are continuous on any rectangle centered at (3,−1) that does not
intersect the x-axis.
5. Two solutions are
1
y =− ln e −2y 0 + 2(x − x 0 )
2
and
1
y =− ln e −2y 0 + 2(x 0 − x) .
2
The theorem does not apply, because the differential equation is y =±2y.

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October 14, 2010 17:50 THM/NEIL Page-802 27410_25_Ans_p801-866