 Calculus Volume 2

# Chapter 4

### Checkpoint

4.2

$5 5$

4.3

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

4.5

$y = 1 3 x 3 − 2 x 2 + 3 x − 6 e x + 14 y = 1 3 x 3 − 2 x 2 + 3 x − 6 e x + 14$

4.6

$v ( t ) = −9.8 t v ( t ) = −9.8 t$

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

$y = 2 + C e x 2 + 3 x y = 2 + C e x 2 + 3 x$

4.11

$y = 4 + 14 e x 2 + x 1 − 7 e x 2 + x y = 4 + 14 e x 2 + x 1 − 7 e x 2 + x$

4.12

Initial value problem:

$d u d t = 2.4 − 2 u 25 , u ( 0 ) = 3 d u d t = 2.4 − 2 u 25 , u ( 0 ) = 3$

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

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

$y = x 3 + x 2 + C x − 2 y = x 3 + x 2 + C x − 2$

4.17

$y = −2 x − 4 + 2 e 2 x y = −2 x − 4 + 2 e 2 x$

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:

$8 q ′ + 1 0.02 q = 20 sin 5 t , q ( 0 ) = 4 8 q ′ + 1 0.02 q = 20 sin 5 t , q ( 0 ) = 4$

$q ( t ) = 10 sin 5 t − 8 cos 5 t + 172 e −6.25 t 41 q ( t ) = 10 sin 5 t − 8 cos 5 t + 172 e −6.25 t 41$

### Section 4.1 Exercises

1 .

$1 1$

3 .

$3 3$

5 .

$1 1$

7 .

$1 1$

19 .

$y = 4 + 3 x 4 4 y = 4 + 3 x 4 4$

21 .

$y = 1 2 e x 2 y = 1 2 e x 2$

23 .

$y = 2 e − 1 / x y = 2 e − 1 / x$

25 .

$u = sin −1 ( e −1 + t ) u = sin −1 ( e −1 + t )$

27 .

$y = − x + 1 1 − x − 1 y = − x + 1 1 − x − 1$

29 .

$y = C − x + x ln x − ln ( cos x ) y = C − x + x ln x − ln ( cos x )$

31 .

$y = C + 4 x ln ( 4 ) y = C + 4 x ln ( 4 )$

33 .

$y = 2 3 t 2 + 16 ( t 2 + 16 ) + C y = 2 3 t 2 + 16 ( t 2 + 16 ) + C$

35 .

$x = 2 15 4 + t ( 3 t 2 + 4 t − 32 ) + C x = 2 15 4 + t ( 3 t 2 + 4 t − 32 ) + C$

37 .

$y = C x y = C x$

39 .

$y = 1 − t 2 2 , y = − t 2 2 − 1 y = 1 − t 2 2 , y = − t 2 2 − 1$

41 .

$y = e − t , y = − e − t y = e − t , y = − e − t$

43 .

$y = 2 ( t 2 + 5 ) , t = 3 5 y = 2 ( t 2 + 5 ) , t = 3 5$

45 .

$y = 10 e −2 t , t = − 1 2 ln ( 1 10 ) y = 10 e −2 t , t = − 1 2 ln ( 1 10 )$

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 .

$v ( t ) = −32 t + a v ( t ) = −32 t + a$

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 .

$y = 4 e 3 t y = 4 e 3 t$

63 .

$y = 3 − 2 t + t 2 y = 3 − 2 t + t 2$

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 .

$y ′ = 2 e t 2 / 2 y ′ = 2 e t 2 / 2$

109 .

$2 2$

111 .

$3.2756 3.2756$

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 .

$4.0741 e −10 4.0741 e −10$

### Section 4.3 Exercises

119 .

$y = e t − 1 y = e t − 1$

121 .

$y = 1 − e − t y = 1 − e − t$

123 .

$y = C x e −1 / x y = C x e −1 / x$

125 .

$y = 1 C − x 2 y = 1 C − x 2$

127 .

$y = − 2 C + ln x y = − 2 C + ln x$

129 .

$y = C e x ( x + 1 ) + 1 y = C e x ( x + 1 ) + 1$

131 .

$y = sin ( ln t + C ) y = sin ( ln t + C )$

133 .

$y = − ln ( e − x ) y = − ln ( e − x )$

135 .

$y = 1 2 − e x 2 y = 1 2 − e x 2$

137 .

$y = tanh −1 ( x 2 2 ) y = tanh −1 ( x 2 2 )$

139 .

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

141 .

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

143 .

$y=Ce−2x+12y=Ce−2x+12$ 145 .

$y=12C−exy=12C−ex$ 147 .

$y=Ce−xxxy=Ce−xxx$ 149 .

$y = r d ( 1 − e − d t ) y = r d ( 1 − e − d t )$

151 .

$y ( t ) = 10 − 9 e − x / 50 y ( t ) = 10 − 9 e − x / 50$

153 .

$134.3134.3$ kilograms

155 .

$720720$ seconds

157 .

$2424$ hours $5757$ minutes

159 .

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

161 .

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

163 .

$y = ( c + b a ) e a x − b a y = ( c + b a ) e a x − b a$

165 .

$y ( t ) = c L + ( I − c L ) e − r t / L y ( t ) = c L + ( I − c L ) e − r t / L$

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 .

$P = 10 e 10 x e 10 x + 4 P = 10 e 10 x e 10 x + 4$

173 .

$P ( t ) = 10000 e 0.02 t 150 + 50 e 0.02 t P ( t ) = 10000 e 0.02 t 150 + 50 e 0.02 t$

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 .

$y = −20 4 × 10 −6 − 0.002 e 0.01 t y = −20 4 × 10 −6 − 0.002 e 0.01 t$

189 . 191 .

$P ( t ) = 850 + 500 e 0.009 t 85 + 5 e 0.009 t P ( t ) = 850 + 500 e 0.009 t 85 + 5 e 0.009 t$

193 .

$1313$ years months

195 . 197 .

$31.46531.465$ days

199 .

September $20082008$

201 .

$K + T 2 K + T 2$

203 .

$r = 0.0405 r = 0.0405$

205 .

$α = 0.0081 α = 0.0081$

207 .

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

### Section 4.5 Exercises

209 .

Yes

211 .

Yes

213 .

$y ′ − x 3 y = sin x y ′ − x 3 y = sin x$

215 .

$y ′ + ( 3 x + 2 ) x y = − e x y ′ + ( 3 x + 2 ) x y = − e x$

217 .

$d y d t − y x ( x + 1 ) = 0 d y d t − y x ( x + 1 ) = 0$

219 .

$e x e x$

221 .

$− ln ( cosh x ) − ln ( cosh x )$

223 .

$y = C e 3 x − 2 3 y = C e 3 x − 2 3$

225 .

$y = C x 3 + 6 x 2 y = C x 3 + 6 x 2$

227 .

$y = C e x 2 / 2 − 3 y = C e x 2 / 2 − 3$

229 .

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

231 .

$y = C x 3 − x 2 y = C x 3 − x 2$

233 .

$y = C ( x + 2 ) 2 + 1 2 y = C ( x + 2 ) 2 + 1 2$

235 .

$y = C x + 2 sin ( 3 t ) y = C x + 2 sin ( 3 t )$

237 .

$y = C ( x + 1 ) 3 − x 2 − 2 x − 1 y = C ( x + 1 ) 3 − x 2 − 2 x − 1$

239 .

$y = C e sinh −1 x − 2 y = C e sinh −1 x − 2$

241 .

$y = x + 4 e x − 1 y = x + 4 e x − 1$

243 .

$y = − 3 x 2 ( x 2 − 1 ) y = − 3 x 2 ( x 2 − 1 )$

245 .

$y = 1 − e tan −1 x y = 1 − e tan −1 x$

247 .

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

249 .

$y = 2 e 2 x − 2 x − 2 x − 1 y = 2 e 2 x − 2 x − 2 x − 1$

251 .

$v ( t ) = g m k ( 1 − e − k t / m ) v ( t ) = g m k ( 1 − e − k t / m )$

253 .

$40.45140.451$ seconds

255 .

$g m k g m k$

257 .

$y = C e x − a ( x + 1 ) y = C e x − a ( x + 1 )$

259 .

$y = C e x 2 / 2 − a y = C e x 2 / 2 − a$

261 .

$y = e k t − e t k − 1 y = e k t − e t k − 1$

### Review Exercises

263 .

F

265 .

T

267 .

$y ( x ) = 2 x ln ( 2 ) + x cos −1 x − 1 − x 2 + C y ( x ) = 2 x ln ( 2 ) + x cos −1 x − 1 − x 2 + C$

269 .

$y ( x ) = ln ( C − cos x ) y ( x ) = ln ( C − cos x )$

271 .

$y ( x ) = e e C + x y ( x ) = e e C + x$

273 .

$y ( x ) = 4 + 3 2 x 2 + 2 x − sin x y ( x ) = 4 + 3 2 x 2 + 2 x − sin x$

275 .

$y ( x ) = − 2 1 + 3 ( x 2 + 2 sin x ) y ( x ) = − 2 1 + 3 ( x 2 + 2 sin x )$

277 .

$y ( x ) = −2 x 2 − 2 x − 1 3 − 2 3 e 3 x y ( x ) = −2 x 2 − 2 x − 1 3 − 2 3 e 3 x$

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 .

$P ( t ) = 1600000 e 0.02 t 9840 + 160 e 0.02 t P ( t ) = 1600000 e 0.02 t 9840 + 160 e 0.02 t$

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