Calculus Volume 2

# Chapter 4

### Checkpoint

4.2

$55$

4.3

$y=2x2+3x+2y=2x2+3x+2$

4.5

$y=13x3−2x2+3x−6ex+14y=13x3−2x2+3x−6ex+14$

4.6

$v(t)=−9.8tv(t)=−9.8t$

4.7 4.8 The equilibrium solutions are $y=−2y=−2$ and $y=2.y=2.$ For this equation, $y=−2y=−2$ is an unstable equilibrium solution, and $y=2y=2$ is a semi-stable equilibrium solution.

4.9
$nn$ $xnxn$ $yn=yn−1+hf(xn−1,yn−1)yn=yn−1+hf(xn−1,yn−1)$
$00$ $11$ $−2−2$
$11$ $1.11.1$ $y1=y0+hf(x0,y0)=−1.5y1=y0+hf(x0,y0)=−1.5$
$22$ $1.21.2$ $y2=y1+hf(x1,y1)=−1.1419y2=y1+hf(x1,y1)=−1.1419$
$33$ $1.31.3$ $y3=y2+hf(x2,y2)=−0.8387y3=y2+hf(x2,y2)=−0.8387$
$44$ $1.41.4$ $y4=y3+hf(x3,y3)=−0.5487y4=y3+hf(x3,y3)=−0.5487$
$55$ $1.51.5$ $y5=y4+hf(x4,y4)=−0.2442y5=y4+hf(x4,y4)=−0.2442$
$66$ $1.61.6$ $y6=y5+hf(x5,y5)=0.0993y6=y5+hf(x5,y5)=0.0993$
$77$ $1.71.7$ $y7=y6+hf(x6,y6)=0.5099y7=y6+hf(x6,y6)=0.5099$
$88$ $1.81.8$ $y8=y7+hf(x7,y7)=1.0272y8=y7+hf(x7,y7)=1.0272$
$99$ $1.91.9$ $y9=y8+hf(x8,y8)=1.7159y9=y8+hf(x8,y8)=1.7159$
$1010$ $22$ $y10=y9+hf(x9,y9)=2.6962y10=y9+hf(x9,y9)=2.6962$
4.10

$y=2+Cex2+3xy=2+Cex2+3x$

4.11

$y=4+14ex2+x1−7ex2+xy=4+14ex2+x1−7ex2+x$

4.12

Initial value problem:

$dudt=2.4−2u25,u(0)=3dudt=2.4−2u25,u(0)=3$

$Solution:u(t)=30−27e−t/50Solution:u(t)=30−27e−t/50$

4.13
1. Initial-value problem
$dTdt=k(T−70),T(0)=450dTdt=k(T−70),T(0)=450$
2. $T(t)=70+380ektT(t)=70+380ekt$
3. Approximately $114114$ minutes.
4.14
1. $dPdt=0.04(1−P750),P(0)=200dPdt=0.04(1−P750),P(0)=200$

2. 3. $P(t)=3000e.04t11+4e.04tP(t)=3000e.04t11+4e.04t$

4. After $1212$ months, the population will be $P(12)≈278P(12)≈278$ rabbits.

4.15

$y′+15x+3y=10x−20x+3;p(x)=15x+3y′+15x+3y=10x−20x+3;p(x)=15x+3$ and $q(x)=10x−20x+3q(x)=10x−20x+3$

4.16

$y=x3+x2+Cx−2y=x3+x2+Cx−2$

4.17

$y=−2x−4+2e2xy=−2x−4+2e2x$

4.18
1. $dvdt=−v−9.8v(0)=0dvdt=−v−9.8v(0)=0$
2. $v(t)=9.8(e−t−1)v(t)=9.8(e−t−1)$
3. $limt→∞v(t)=limt→∞(9.8(e−t−1))=−9.8m/s≈−21.922mphlimt→∞v(t)=limt→∞(9.8(e−t−1))=−9.8m/s≈−21.922mph$
4.19

Initial-value problem:

$8q′+10.02q=20sin5t,q(0)=48q′+10.02q=20sin5t,q(0)=4$

$q(t)=10sin5t−8cos5t+172e−6.25t41q(t)=10sin5t−8cos5t+172e−6.25t41$

### Section 4.1 Exercises

1.

$11$

3.

$33$

5.

$11$

7.

$11$

19.

$y=4+3x44y=4+3x44$

21.

$y=12ex2y=12ex2$

23.

$y=2e−1/xy=2e−1/x$

25.

$u=sin−1(e−1+t)u=sin−1(e−1+t)$

27.

$y=−x+11−x−1y=−x+11−x−1$

29.

$y=C−x+xlnx−ln(cosx)y=C−x+xlnx−ln(cosx)$

31.

$y=C+4xln(4)y=C+4xln(4)$

33.

$y=23t2+16(t2+16)+Cy=23t2+16(t2+16)+C$

35.

$x=2154+t(3t2+4t−32)+Cx=2154+t(3t2+4t−32)+C$

37.

$y=Cxy=Cx$

39.

$y=1−t22,y=−t22−1y=1−t22,y=−t22−1$

41.

$y=e−t,y=−e−ty=e−t,y=−e−t$

43.

$y=2(t2+5),t=35y=2(t2+5),t=35$

45.

$y=10e−2t,t=−12ln(110)y=10e−2t,t=−12ln(110)$

47.

$y=14(41−e−4t),y=14(41−e−4t),$ never

49.

Solution changes from increasing to decreasing at $y(0)=0y(0)=0$

51.

Solution changes from increasing to decreasing at $y(0)=0y(0)=0$

53.

$v(t)=−32t+av(t)=−32t+a$

55.

$00$ ft/s

57.

$4.864.86$ meters

59.

$x=50t−15π2cos(πt)+3π2,2x=50t−15π2cos(πt)+3π2,2$ hours $11$ minute

61.

$y=4e3ty=4e3t$

63.

$y=3−2t+t2y=3−2t+t2$

65.

$y=1k(ekt−1)y=1k(ekt−1)$ and $y=xy=x$

### Section 4.2 Exercises

67. 69.

$y=0y=0$ is a stable equilibrium

71. 73.

$y=0y=0$ is a stable equilibrium and $y=2y=2$ is unstable

75. 77. 79. 81. 83. 85.

E

87.

A

89.

B

91.

A

93.

C

95.

$2.24,2.24,$ exact: $33$

97.

$7.739364,7.739364,$ exact: $5(e−1)5(e−1)$

99.

$−0.2535−0.2535$ exact: $00$

101.

$1.345,1.345,$ exact: $1ln(2)1ln(2)$

103.

$−4,−4,$ exact: $−1/2−1/2$

105. 107.

$y′=2et2/2y′=2et2/2$

109.

$22$

111.

$3.27563.2756$

113.

$2e2e$

Step Size Error
$h=1h=1$ $0.39350.3935$
$h=10h=10$ $0.061630.06163$
$h=100h=100$ $0.0066120.006612$
$h=1000h=1000$ $0.00066610.0006661$
115. 117.

$4.0741e−104.0741e−10$

### Section 4.3 Exercises

119.

$y=et−1y=et−1$

121.

$y=1−e−ty=1−e−t$

123.

$y=Cxe−1/xy=Cxe−1/x$

125.

$y=1C−x2y=1C−x2$

127.

$y=−2C+lnxy=−2C+lnx$

129.

$y=Cex(x+1)+1y=Cex(x+1)+1$

131.

$y=sin(lnt+C)y=sin(lnt+C)$

133.

$y=−ln(e−x)y=−ln(e−x)$

135.

$y=12−ex2y=12−ex2$

137.

$y=tanh−1(x22)y=tanh−1(x22)$

139.

$x=sin(1-t+tlnt)x=sin(1-t+tlnt)$

141.

$y=ln(ln(5))−ln(2−5x)y=ln(ln(5))−ln(2−5x)$

143.

$y=Ce−2x+12y=Ce−2x+12$ 145.

$y=12C−exy=12C−ex$ 147.

$y=Ce−xxxy=Ce−xxx$ 149.

$y=rd(1−e−dt)y=rd(1−e−dt)$

151.

$y(t)=10−9e−x/50y(t)=10−9e−x/50$

153.

$134.3134.3$ kilograms

155.

$720720$ seconds

157.

$2424$ hours $5757$ minutes

159.

$T(t)=20+50e−0.125tT(t)=20+50e−0.125t$

161.

$T(t)=20+38.5e−0.125tT(t)=20+38.5e−0.125t$

163.

$y=(c+ba)eax−bay=(c+ba)eax−ba$

165.

$y(t)=cL+(I−cL)e−rt/Ly(t)=cL+(I−cL)e−rt/L$

167.

$y=40(1−e−0.1t),40y=40(1−e−0.1t),40$ g/cm2

### Section 4.4 Exercises

169. $P=0P=0$ semi-stable

171.

$P=10e10xe10x+4P=10e10xe10x+4$

173.

$P(t)=10000e0.02t150+50e0.02tP(t)=10000e0.02t150+50e0.02t$

175.

$6969$ hours $55$ minutes

177.

$88$ years $1111$ months

179. 181. $P1P1$ semi-stable

183. $P2>0P2>0$ stable

185. $P1=0P1=0$ is semi-stable

187.

$y=−204×10−6−0.002e0.01ty=−204×10−6−0.002e0.01t$

189. 191.

$P(t)=850+500e0.009t85+5e0.009tP(t)=850+500e0.009t85+5e0.009t$

193.

$1313$ years months

195. 197.

$31.46531.465$ days

199.

September $20082008$

201.

$K+T2K+T2$

203.

$r=0.0405r=0.0405$

205.

$α=0.0081α=0.0081$

207.

Logistic: $361,361,$ Threshold: $436,436,$ Gompertz: $309.309.$

### Section 4.5 Exercises

209.

Yes

211.

Yes

213.

$y′−x3y=sinxy′−x3y=sinx$

215.

$y′+(3x+2)xy=−exy′+(3x+2)xy=−ex$

217.

$dydt−yx(x+1)=0dydt−yx(x+1)=0$

219.

$exex$

221.

$−ln(coshx)−ln(coshx)$

223.

$y=Ce3x−23y=Ce3x−23$

225.

$y=Cx3+6x2y=Cx3+6x2$

227.

$y=Cex2/2−3y=Cex2/2−3$

229.

$y=Ctan(x2)−2x+4tan(x2)ln(sin(x2))y=Ctan(x2)−2x+4tan(x2)ln(sin(x2))$

231.

$y=Cx3−x2y=Cx3−x2$

233.

$y=C(x+2)2+12y=C(x+2)2+12$

235.

$y=Cx+2sin(3t)y=Cx+2sin(3t)$

237.

$y=C(x+1)3−x2−2x−1y=C(x+1)3−x2−2x−1$

239.

$y=Cesinh−1x−2y=Cesinh−1x−2$

241.

$y=x+4ex−1y=x+4ex−1$

243.

$y=−3x2(x2−1)y=−3x2(x2−1)$

245.

$y=1−etan−1xy=1−etan−1x$

247.

$y=(x+2)ln(x+22)y=(x+2)ln(x+22)$

249.

$y=2e2x−2x−2x−1y=2e2x−2x−2x−1$

251.

$v(t)=gmk(1−e−kt/m)v(t)=gmk(1−e−kt/m)$

253.

$40.45140.451$ seconds

255.

$gmkgmk$

257.

$y=Cex−a(x+1)y=Cex−a(x+1)$

259.

$y=Cex2/2−ay=Cex2/2−a$

261.

$y=ekt−etk−1y=ekt−etk−1$

### Chapter Review Exercises

263.

F

265.

T

267.

$y(x)=2xln(2)+xcos−1x−1−x2+Cy(x)=2xln(2)+xcos−1x−1−x2+C$

269.

$y(x)=ln(C−cosx)y(x)=ln(C−cosx)$

271.

$y(x)=eeC+xy(x)=eeC+x$

273.

$y(x)=4+32x2+2x−sinxy(x)=4+32x2+2x−sinx$

275.

$y(x)=−21+3(x2+2sinx)y(x)=−21+3(x2+2sinx)$

277.

$y(x)=−2x2−2x−13−23e3xy(x)=−2x2−2x−13−23e3x$

279. $y(x)=Ce−x+lnxy(x)=Ce−x+lnx$

281.

Euler: $0.6939,0.6939,$ exact solution: $y(x)=3x−e−2x2+ln(3)y(x)=3x−e−2x2+ln(3)$

283.

$40494049$ second

285.

$x(t)=5000+2459−493t−2459e−5/3t,t=307.8x(t)=5000+2459−493t−2459e−5/3t,t=307.8$ seconds

287.

$T(t)=200(1−e−t/1000)T(t)=200(1−e−t/1000)$

289.

$P(t)=1600000e0.02t9840+160e0.02tP(t)=1600000e0.02t9840+160e0.02t$