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

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
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21. (a) 50.408 m (b) 5.248 sec 22. (a) 45.783 m (b) 5.129 sec
23. (a) 48.562 m (b) 5.194 sec
24. (a) 176.7 ft/sec (b) 1074.5 ft (c) 15 ft/sec (d) 256.6 sec
25. (a).
ln 1 +
kvr
+
P = T ln 1 +
k vr
k2 V mg) k m k \ mg 2
v = -(mg/k) + [v0 + (mg/k)] exp(kt/m) (b) v = v0  gt; yes
v = 0 for t > 0
vL = 2a2g(p  p')/9g (b) e = 4n a3g(p  p')/3 E
11.58 m/sec (b) 13.45m (c) k > 0.2394 kg/sec
29. (a) v = R^2g/(R + x) (b) 50.6 hr
(1+1 )"'?
31. (b) x = utcos A, y = gt2/2 + ut sin A + h
(d) 16L2/(u2 cos2 A) + L tan A + 3 > H
(f) u = 106.89 ft/sec, A = 0.7954 rad
32. (a) v = (u cos A)ert, w = g/r + (u sin A + g/r)e-rt x = u cos A(1  ert)/r, y = -gt/r + (u sin A + g/r)(1  ert)/r + h
u 145.3 ft/sec, A 0.644 rad
26. (a)
(c)
27. (a)
28. (a)
30. v =
altitude = 1536 miles
(b)
(d)
33. (d) k = 2.193
Section 2.4, page 72
14
15
2. 0 < t < 4 4. to < t < 2 6. 1 < t < n 8. t2 + y2 < 1 t = 0, y = 0
11. y = 0, y = 3
1. 0 < t < 3
3. n/2 < t < 3n/2
5. -2 < t < 2
7. 2t + 5y > 0or2t + 5y < 0
9. 1  t2 + y2 > 0 or 1  t2 + y2 < 0,
10. Everywhere
12. t = an for n = 0, ±1, ±2,...; 13. y = ±^>0  4Z2 if y0 = 0;
y =1 |t| < |y0|/2
y = [(1/y0)  t2]-1 if y0 = 0; y = 0 if y0 = 0; interval is |t| < 1/^yj
if y0 > 0; to < t < to if y0 < 0
y = yJ^y+X if y0 = 0; y = 0 if y0 = 0; interval is  1 /2y/ < t < to if y0 = 0; to < t < to if y0 = 0
y = ±§ ln(1 + t3) + y2; [1  exp(3^/2)]1/3 < t < to
17. y ^ 3 if y0 > 0; y = 0 if y0 = 0; y to if y0 < 0
18. y ^ to if y0 < 0; y ^ 0 if y0 > 0 19. y ^ 0 if y, < 9; y-
20. y ^ to if y0 < yc ~ 0.019; otherwise y is asymptotic to Vt  1
21. (a) No (b) Yes; set t0 = 1/2 in Eq. (19) in text.
(c) |y| < (4/3)3/2 = 1.5396
22. (a) y1(t) is a solution for t > 2; y2(t) is a solution for all t
(b) f is not continuous at (2, 1)
16. y
to if w > 9
0
684
See SSM for detailed solutions to 23a, 24, 25, 27ab
28, 29, 32
3, 6, 7c, 9
11, 14, 16bc, 17a, 18abc
20bd, 21abd, 24abc, 26ab
3
5, 7, 9, 12, 14
15
1 1 rt
26. (a) Yi(t) =  ; y2(f) =  *(s)g(s) ds
1 *(t) ?(t)Jt0
28. y =±[5t/(2 + 5ct5)]1/2 29. y = r/(k + cre-rt)
30. y =±[f/(o + cee-2et)]1/2 I ,T , t 1)1/2
31. y
 y =± *(t)/ 21 *(s) ds + c
, where *(t) = exp(2T sin t + 2Tt)
32. / = 1 (1 - e-2t) for 0 < t < 1; y = 1 (e2 - 1)e-2t for t > 1
33. y = e-2t for 0 < t < 1; y = e-(t+1) for t > 1
Section 2.5, page 84
1. y = 0 is unstable
2. y = -a/b is asymptotically stable, y = 0 is unstable
3. y = 1 is asymptotically stable, y = 0 and y = 2 are unstable
4. y = 0 is unstable 5. y = 0 is asymptotically stable
6. y = 0 is asymptotically stable
7. (c) y = [y + (1 - y0)kt]/[1 + (1 - y^kt]
8. y = 1 is semistable
9. y =  1 is asymptotically stable, y = 0 is semistable, y = 1 is unstable
10. y = -1 and y = 1 are asymptotically stable, y = 0 is unstable
11. y = 0 is asymptotically stable, y = b2/a2 is unstable
12. y = 2 is asymptotically stable, y = 0 is semistable, y = 2 is unstable
13. y = 0 and y = 1 are semistable a) t = (1/r) ln4; 55.452 years (b) T = (1 /r) ln[?(1 - a)/(1 - fi)a]; 75.78 years
a) y = 0 is unstable, y = K is asymptotically stable
b) Concave up for 0 < y < K/ e, concave down for K/e < y < K
a) y = Kexp{[ln(y0/K)]e-rt} (b) y(2) = 0.7153K = 57.6 x 106 kg
c) t = 2.215 years
b) (h/a)^k/an ; yes 19. (b) k2/2g(aa)2
c) k/a < n a2
c) Y = Ey2 = KE[1 - (E/r)] (d) Ym = Kr/4 for E = r/2
a) yu = K[1 T V1 - (4h/rK)]/2
a) y = 0 is unstable, y = 1 is asymptotically stable
b) y = y0/[y0 + (1 - y0)erat]
a) y = y{)e^^ (b) v = ^0 exp[-ay0(1 - e~fit)/0] (c) ^ exp(-ay0/^)
b) z = 1/[v + (1 - v)e?] (c) 0.0927
pq[ea(q-p)t - 1]
a) lim x(t) = min(p, q); x(t) =
15.
16.
17.
18.
20.
21.
22.
23.
24.
26.
b) lim x (t) = p; x (t) =
2
p at
t^-? v ' pat + 1
Section 2.6, page 95
qea(q-p)t - p
1. x2 + 3x + y2 - 2y = c 2. Not exact
3. x3 - x2y + 2x + 2t3 + 3y = c 4. x2 y2 + 2xy = c
5. ax2 + 2bxy + cy2 = k 6. Not exact
7. ex sin y + 2y cos x = c; also y = 0 8. Not exact
9. exy cos 2x + x2 - 3y = c 10. ylnx + 3X2 - 2y = c
11. Not exact 12. x2 + y2 = c
13. y = [x + \/28 - 3x2]/2, |x| < V28/3
14. y = [x - (24a3 + x2 - 8x - 16)1/2]/4, x > 0.9846
15. b = 3; x2 y2 + 2a3 y = c 16. b = 1; e2xy + x2 = c
See SSM for detailed solutions
to 19, 22, 23
25, 26, 27, 29, 31
1d, 3a
3bcd, 4d, 6, 9
13a, 15ac, 16
17. y N(x, y) dy + f [M(x, y)  f Nx(x, y) dy] dx
19. x2 + 2 ln |y| y2 = c; also y = 0 20. ex siny + 2ycos x = c
21. xy2  (y2  2y + 2)ey = c 22. x2ex siny = c
24. /x(t) = expyR(t) dt, where t = xy
25. ^(x) = e3x; (3x2y + y3)e3x = c 26. ^(x) = ex; y = cex + 1 + e2x
27. n.(y) = y; xy + ycosy  siny = c
28. n(y) = e2y/y; xe2y  ln |y| = c; also y = 0
29. ^(y) = siny; ex siny + y2 = c 30. ^(y) = y2; x4 + 3xy + y4 = c
31. ^(x, y) = xy; x3y + 3x2 + y3 = c
Section 2.7, page 103
1. (a) 1.2, 1.39, 1.571, 1.7439
(b) 1.1975, 1.38549, 1.56491, 1.73658
(c) 1.19631, 1.38335, 1.56200, 1.73308
(d) 1.19516, 1.38127, 1.55918, 1.72968
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