Books
in black and white
Main menu
Share a book About us Home
Books
Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics
Ads

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
Download (direct link): elementarydifferentialequat2001.pdf
Previous << 1 .. 291 292 293 294 295 296 < 297 > 298 299 300 301 302 303 .. 486 >> Next

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
(e) 0.63 rad < A < 0.96 rad
(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
Answers to Problems
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
Answers to Problems
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
Previous << 1 .. 291 292 293 294 295 296 < 297 > 298 299 300 301 302 303 .. 486 >> Next