Transcription of Answers to Selected Exercises - University of Alabama in ...
1 Answers to Selected ExercisesChapter , fifth, fifth, forty-second3a , it is3a , it is not3a +95c +56b 36b 1 37b cos(2x)+4 sin(2x)7b (2x)8a , because then 7=y (1)=4 .8a x2=3xd yd x+3y+2x,y (1)=298a (0)=4 andy (0)=248b 140meters/second9c (0)=0 is replaced withy (0)=29c ii. +2t+10009c iii. 2 4901 (v 2)10b 10b ii.(vhit+ ) +5xln|x| (x)is not continuous atx=0 .Chapter , it is directly , it is not directly +c3b. 5e 4x+ ln|x| 2 x+ x+4+ x2 + (x)+cos(x)+ x2 9 + x 3x+3 + 19x+c3j. 14sin(2x)+c1x+ +32x2+c1x+ +c1x3+c2x2+c3x+ +5e2x 1 for <x< (x+6)2/3+4 for <x< 2 ln|x+1|+8 for 1<x< x 2 ln|x|+4 for 0<x< 4e.
2 Ln|cos(x)|+3 for 2<x< (x)+3 for <x< 4g. 2xln|x|+43 x3+6x+23for 0<x< 5a. 2 cos x2 +2+y(0)5b (x+3)3/2+y(1) 166b to Selected Exercises6b iii. 12e +px2+ 27d. 6erf(3x)+ (x)+ x2 )9a.(0 ifx<0xif 0 x)9b.(2ifx<1x+1 if 1 x)9c. 0ifx<1x 1 if 1 x 21if 2<x 9d. 2x 12x2+cifx 212x2 2s+4+cif 2<x s Chapter yd x=6x 3x y, nonautonomous, const. solns.:y= yd x=sin(x+y)y, nonautonomous, const. solns.: yd x=y3+8 , autonomous, const. solns.:y= yd x=1 y2x, nonautonomous, const. solns.:y=1 andy= yd x=x+y2, nonautonomous, const. solns.: yd x=25y2 y3, autonomous, const. solns.:y=0 ,y=5 andy= yd x=y+3x 2, nonautonomous, const.
3 Solns.:y= yd x=x 3y 2, nonautonomous, const. solns.: yd x=y2 2y 2 , autonomous, const. solns.:y=1+ 3 andy=1 yd x=y2+(x 8)y 8x, nonautonomous, const. solns.:y= F/ ywithF(x,y)=2 yis not continuous at the point(1,0). + +x2+ +x2+14x4+ +9x+ +9x+28x2+29x3+92x4+15x5 Chapter yd x=(3 sin(x)) yd x=4(2 y) yd x=x(4 y) yd x=(x 2)(y 3) yd x=exy 1e px2+ tan(3x+c) pAx2 (c+arctan(x)) (c cos(x)) 32e2x+c + x p5ex2 2x3/2+3 2 1 andy= , , 2 , 3 , .. constant andy= +Aexp 12x2 c 3x cos(x) 1andy=0 Answers to Selected +Aexp 12x2 2x Aex 2y2=2x3+4x+ (arctan(x)+c) :y+y 1=2x2+c1, explicit:y=x2 c2 q c2 x2 2 1 ; alsoy= |x+c| Aex 1 c 3x2 1/2andy= +y3/2=3x+x3/2+ p1+ x3+c + + p18 2 cos(x) (2 x) 2 ln|y|=4x2+ p1+ andy= / Ae xandy=010a.
4 ( , )10b.( ,2)10c.( , )10d.( e, )10e. 2 3,2 3 Chapter yd x+3y=x 2sin(x) yd x (3+x)y= yd x 4y= yd x+0 y= yd x sin(x)y= yd x 827x 1y= x 1cos x2 +ce +ce 4x +cx 32f.[c cosx]x 25 x2h.[x+c]cos(x) 2 4+ce 5x +ce 23b. e 3x e 5x +6x [sin(x) 1] +x2+3 5 1+x2 1 3x2 4+Zx0e3s2sin(s)ds 10+Zx2sin(s) sds 83+Zx3e s2ds 5a. (x)= (x)p(x)andy (x)=f(x) p(x) (x) (x0)y0+Zxx0 (s)f(s)ds Chapter 61a +3y+2=tan(3y+C)1a 12pAe 4x 11a 34+14arcsin(8x+c) 1x 42a x(ln|x|+c) 1andy=02a xpc+2 ln|x|2a (Ax 1) x+p2x2+93a 1+ce6x 1/2andy=03a C x2 1andy=03a sin(x)+csin(x) 3andy= ;y=x(3 ln|x|+c)1 +3y+4 ;y=13 12x+c 2 23x 43andy= 13(2x+4) ;y=(x+c/x)2andy= y;y=4x arccos(x+c) x;y+ln|1 y+x|=candy=x+ ;yln|y| cy=xandy=0618 Answers to Selected ;y= x pln|x|+ 2;y= cx2 2x3 1/2andy= +y 3 ;y=(x+c)2 2x+3 andy=3 +y 3 ;p2x+y 3 ln 1+p2x+y 3 =x+ ;y=x 12ln|x|+c 2 xandy= ;y= 8e2x+ce 12x 1 y+3.
5 Y=3+x+ Ae2x 1 Ae2x+1 1andy=x+ +x2;y=c2+2cxandy= (y);y=arcsin (c+x)e x 2;y=x2tan(c+ln|x|) x+(1 n)p(x)u=(1 n)f(x)Chapter +3xd yd x=0 ,y=cx1b. 6x2y+ 2y 2x3 d yd x=0 ,y=x3 pc+x61c. 2x y y3 + x2 3x y2 d yd x=0 ,x2y x y3= (y)+x1+y2d yd x=0 ,y=tan cx qx+cx4a. (x,y)=x2y+x y2,y= x2 12x x2+C4b. (x,y)=x2y3+x4,y= cx 2 x2 1/34c. (x,y)=2x x2+y3,y= c+x3 2x 1/34d. (x,y)=x3y2+x+3y2,y= rc xx3+34e. (x,y)=x4y 15y5,x4y 15y5=c4f. (x,y)=xln|x y|,y=x 1ecx 14g. (x,y)=x+x ey,y=ln c xx 4h. (x,y)=x ey+y,x ey+y=c5a. =x 3,y4 x 2y3=c5b. =y 4,x y 3+y=c5c. =x 2y2,y4 x 2y3=c5d. =cos(y),xcos(y)+sin(y)=c5e. = x,y= 12 px2+C x2x5f.
6 =e x2,y=C ex2 15g. =x3,y= cx 4 1 1/45h. = y,y5/2+x2y3/2=c5i. =x y1/3,x4y7/3+x2y1/3=cChapter (0)should be within1/6of a., (8) 1/2 Answers to Selected Exercises6194. a., (4) 31/2, (4) 4 , (4) 61/35. ai. & bi., max. is approximately 61/2and is atx 3 ., (10) 21/2, (10) 13/46. ai. & bi., max. is approximately 5 and is atx 2 . , (10) 21/2, max. is approximately 3 and is atx 61/2. , (10) 21/27. ai. & bi., (3) 31/2, (0) to be an asymptotically stable constant appears to be anunstable constant appears to be a stable (but maybe notasymptotically stable) constant appears to be an unstable appears to be an unstable constant solution, andy= 2 appears tobe an asymptotically stable constant appears to be a stable (but notasymptotically stable) constant to Selected ExercisesChapter 114/3 4/325/3 5/332 (3) with error= (3) with error=.
7 5976 (3) with error=.03097 (3) with error=.1561 (3) with error=.0125 (3) with error=.0013 (3) with error=.0001 (x)=1 2323x+10,y(4)=791026c (4) with error= (4) with error= (4) with error= (4) with error= ( 1)k37a does (x) 0 asx ;|yk y(xk)| 37b 35 k7b solution becomes are nonsense. The solution to the initial-value problem is not valid forx>7 ,538, each, the answer is ln 2( ) each, the answer is ln 10( ) (t)=e twith =110ln(50) , grams5b grams5b %6b %6b %6b %6b % Answers to Selected Exercises6216b % ,953 years ,222 years ago6e.
8 5730ln 2ln AA0 7b rabbits7b ,953, rabbits7b ,999, rabbits7c. =R0R(t)1 e tR0 R(t)e t7d , rabbits7d , Rdt=54R is the equilibrium solution. If we start with more,the population increases. If we start with less, the population decreases (rapidly!) (t)=400+(R0 400)e5 Rdt= is no equilibriumsolution; if we start with a positive number of rabbits, the population constantly (t)=R0e3 Rdt= R R2 Rdt= 14 R ydt=50 120y(initial condition:y(0)=200 ) (t)=1000 800e ln 8 weeks (about weeks) ydt=94 750e 3t/100012d gallons12d gallons12d 3 ydt=94 650e 3t/100013d gallons13d gallons13d ydt=32 1400y14a (t)=600 600e t/40014b oz.
9 Salt/gal. ln 3 ydt=32 1200y15a (t)=300 300e t/20015b salt/gal. t16b ydt=4 3500 ty16b (500 t) h10 t50i316c the butler. The time of death wasonly about about 3:20, over an hour after the butler reportedfinding the body. Besides, hisfingerprints were on the +c1x3+ + + +c1e 2x+ x2+ (x)+ +23c1x3+c12x+ 13(2x+c1)3 |1+Ax|+C,y=candy=x+ +c1x2+ +Ae2x+ 3e 3x+Ae 4x+ ln A+e 2x +B x+ Aln|x|+B x+ (x+A)4+B x+ 2x+B x2+C x+ (Ax+c) (c Ax) + pAe2x+ +3y=Ax+ Ax+ 13(2x+c1)3 +1 Aandy= +Ae2x+ + ln A+Be 2x 622 Answers to Selected Exercises6a +2x3+56a +86a +36a +e 2x 26a +3x+56a 2 ln|x|+2x+136b 2x+36b (2x+1)3/2 37a (x+4)37a (3x 2)37b +57b 2ex 1 8a arctan(2x)+38a 1x8a x 2x+2 + (Ax)+c,y=c 1xandy=12aln x ax+a +c9a (x)9a x9a e2x1+ (Ax+B)
10 ,y=1c xandy=c1+Be2cx1 Be2cxChapter , linear, , linear, , linear, , , linear, , linear, , , linear, , linear, , , linear, , linear, x2+5dd x+63b sin(x)+5 cos(x)3b +10x+ x2 5dd x+94b sin(x) 5 cos(x)4b ii. 15 cos(3x)4b iv.[2 sin(x) cos(x)] x2+5xdd x+65b sin(x)+5xcos(x) x2sin(x)5b ii. 16x2+20x+6 e4x5b x3 sin(x)dd x+cos(x)6b i. cos(x)6b +sin(x)6b (x) 2xsin(x) (x)=2 cos(2x)+3 sin(2x) (x)=3e2x 3e (x)=3e2x+5e (x)=e2x+4x (x)=5x1/2+3x 1 (x)=5x 2xln|x| (x)= 3 cos x2 2 sin x2 (x)=4x2+ (x)=4 cos(2x)+4 sin(2x) (x)=3+2 sin2(x)+8 sin(x)cos(x) (x)=2 sin(x)+2 sinh(x) x2 x2 1 ,L1L2=d2d x2 x2+1 x2+ x2+x3 dd x+ 2x+x5 ,L1L2=d2d x2+ x2+x3 dd x+ 3x2+x5 x2+ 4+2x2 dd x+6x,L1L2=xd2d x2+ 3+2x2 dd x+ x2,L1L2=xd2d x2+2dd x2,L1L2=x3d2d x2+6x2dd x+ (x)d2d x2,L1L2=sin(x)d2d x2+2 cos(x)dd x sin(x)
