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Separable EXERCISES 2.2 Answers to selected odd-numbered problems begin on page ANS-1. In Problems 1-22 solve the given differential equation by separation of variables. 1. \( \frac{d y}{d x}=\sin 5 x \) 2. \( \frac{d y}{d x}=(x+1)^{2} \) 3. \( d x+e^{3 x} d y=0 \) 4. \( d y-(y-1)^{2} d x=0 \) 5. \( x \frac{d y}{d x}=4 y \) 6. \( \frac{d y}{d x}+2 x y^{2}=0 \) 7. \( \frac{d y}{d x}=e^{3 x+2 y} \) 8. \( e^{x} y \frac{d y}{d x}=e^{-y}+e^{-2 x} \), 9. \( y \ln x \frac{d x}{d y}=\left(\frac{y+1}{x}\right)^{2} \) 10. \( \frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2} \) 11. \( \csc y d x+\sec ^{2} x d y=0 \) 12. \( \sin 3 x d x+2 y \cos ^{3} 3 x d y=0 \) 13. \( \left(e^{y}+1\right)^{2} e^{-x} d x+\left(e^{x}+1\right)^{3} e^{-x} d y=0 \) 14. \( x\left(1+y^{2}\right)^{1 / 2} d x=y\left(1+x^{2}\right)^{1 / 2} d y \) 15. \( \frac{d S}{d r}=k S \) 16. \( \frac{d Q}{d t}=k(Q-70) \) 17. \( \frac{d P}{d t}=P-P^{2} \) 18. \( \frac{d N}{d t}+N=N t e^{t+2} \) 19. \( \frac{d y}{d x}=\frac{x y+3 x-y-3}{x y-2 x+4 y-8} \) 20. \( \frac{d y}{d x}=\frac{x y+2 y-x-2}{x y-3 y+x-3} \) 21. \( \frac{d y}{d x}=x \sqrt{1-y^{2}} \) 22. \( \left(e^{x}+e^{-x}\right) \frac{d y}{d x}=y^{2} \) In Problems 23-28 find an explicit solution of the given initial-value problem. 23. \( \frac{d x}{d t}=4\left(x^{2}+1\right), \quad x(\pi / 4)=1 \) 24. \( \frac{d y}{d x}=\frac{y^{2}-1}{x^{2}-1}, \quad y(2)=2 \) 25. \( x^{2} \frac{d y}{d x}=y-x y, \quad y(-1)=-1 \) 26. \( \frac{d y}{d t}+2 y=1, \quad y(0)=\frac{5}{2} \) 27. \( \sqrt{1-y^{2}} d x-\sqrt{1-x^{2}} d y=0, \quad y(0)=\frac{\sqrt{3}}{2} \) 28. \( \left(1+x^{4}\right) d y+x\left(1+4 y^{2}\right) d x=0, \quad y(1)=0 \) Separable - Answers EXERCISES 2.2 (PAGE 52) 1. \( y=-\frac{1}{5} \cos 5 x+c \) 3. \( y=\frac{1}{3} e^{-3 x}+c \) 5. \( y=c x^{4} \) 7. \( -3 e^{-2 y}=2 e^{3 x}+c \) 9. \( \frac{1}{3} x^{3} \ln x-\frac{1}{9} x^{3}=\frac{1}{2} y^{2}+2 y+\ln |y|+c \) 11. \( 4 \cos y=2 x+\sin 2 x+c \) 13. \( \left(e^{x}+1\right)^{-2}+2\left(e^{y}+1\right)^{-1}=c \) 15. \( S=c e^{k r} \) 17. \( P=\frac{c e^{t}}{1+c e^{t}} \) 19. \( (y+3)^{5} e^{x}=c(x+4)^{5} e^{y} \) 21. \( y=\sin \left(\frac{1}{2} x^{2}+c\right) \) 23. \( x=\tan \left(4 t-\frac{3}{4} \pi\right) \) 25. \( y=\frac{e^{-(1+1 / x)}}{x} \) 27. \( y=\frac{1}{2} x+\frac{\sqrt{5}}{2} \sqrt{1-x^{2}} \) 29. \( y=e^{\int i e^{-t^{2}} d t} \)

          Separable
EXERCISES 2.2
Answers to selected odd-numbered problems begin on page ANS-1.

In Problems 1-22 solve the given differential equation by separation of variables.
1. \( \frac{d y}{d x}=\sin 5 x \)
2. \( \frac{d y}{d x}=(x+1)^{2} \)
3. \( d x+e^{3 x} d y=0 \)
4. \( d y-(y-1)^{2} d x=0 \)
5. \( x \frac{d y}{d x}=4 y \)
6. \( \frac{d y}{d x}+2 x y^{2}=0 \)
7. \( \frac{d y}{d x}=e^{3 x+2 y} \)
8. \( e^{x} y \frac{d y}{d x}=e^{-y}+e^{-2 x} \),
9. \( y \ln x \frac{d x}{d y}=\left(\frac{y+1}{x}\right)^{2} \)
10. \( \frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2} \)
11. \( \csc y d x+\sec ^{2} x d y=0 \)
12. \( \sin 3 x d x+2 y \cos ^{3} 3 x d y=0 \)
13. \( \left(e^{y}+1\right)^{2} e^{-x} d x+\left(e^{x}+1\right)^{3} e^{-x} d y=0 \)
14. \( x\left(1+y^{2}\right)^{1 / 2} d x=y\left(1+x^{2}\right)^{1 / 2} d y \)
15. \( \frac{d S}{d r}=k S \)
16. \( \frac{d Q}{d t}=k(Q-70) \)
17. \( \frac{d P}{d t}=P-P^{2} \)
18. \( \frac{d N}{d t}+N=N t e^{t+2} \)
19. \( \frac{d y}{d x}=\frac{x y+3 x-y-3}{x y-2 x+4 y-8} \)
20. \( \frac{d y}{d x}=\frac{x y+2 y-x-2}{x y-3 y+x-3} \)
21. \( \frac{d y}{d x}=x \sqrt{1-y^{2}} \)
22. \( \left(e^{x}+e^{-x}\right) \frac{d y}{d x}=y^{2} \)

In Problems 23-28 find an explicit solution of the given initial-value problem.
23. \( \frac{d x}{d t}=4\left(x^{2}+1\right), \quad x(\pi / 4)=1 \)
24. \( \frac{d y}{d x}=\frac{y^{2}-1}{x^{2}-1}, \quad y(2)=2 \)
25. \( x^{2} \frac{d y}{d x}=y-x y, \quad y(-1)=-1 \)
26. \( \frac{d y}{d t}+2 y=1, \quad y(0)=\frac{5}{2} \)
27. \( \sqrt{1-y^{2}} d x-\sqrt{1-x^{2}} d y=0, \quad y(0)=\frac{\sqrt{3}}{2} \)
28. \( \left(1+x^{4}\right) d y+x\left(1+4 y^{2}\right) d x=0, \quad y(1)=0 \)

Separable - Answers

EXERCISES 2.2 (PAGE 52)
1. \( y=-\frac{1}{5} \cos 5 x+c \)
3. \( y=\frac{1}{3} e^{-3 x}+c \)
5. \( y=c x^{4} \)
7. \( -3 e^{-2 y}=2 e^{3 x}+c \)
9. \( \frac{1}{3} x^{3} \ln x-\frac{1}{9} x^{3}=\frac{1}{2} y^{2}+2 y+\ln |y|+c \)
11. \( 4 \cos y=2 x+\sin 2 x+c \)
13. \( \left(e^{x}+1\right)^{-2}+2\left(e^{y}+1\right)^{-1}=c \)
15. \( S=c e^{k r} \)
17. \( P=\frac{c e^{t}}{1+c e^{t}} \)
19. \( (y+3)^{5} e^{x}=c(x+4)^{5} e^{y} \)
21. \( y=\sin \left(\frac{1}{2} x^{2}+c\right) \)
23. \( x=\tan \left(4 t-\frac{3}{4} \pi\right) \)
25. \( y=\frac{e^{-(1+1 / x)}}{x} \)
27. \( y=\frac{1}{2} x+\frac{\sqrt{5}}{2} \sqrt{1-x^{2}} \)
29. \( y=e^{\int i e^{-t^{2}} d t} \)

Separable
EXERCISES 2.2
Answers to selected odd-numbered problems begin on page ANS-1.

In Problems 1-22 solve the given differential equation by separation of variables.
1. (d y)/(d x)=sin 5 x
2. (d y)/(d x)=(x+1)^2
3. d x+e^3 x d y=0
4. d y-(y-1)^2 d x=0
5. x (d y)/(d x)=4 y
6. (d y)/(d x)+2 x y^2=0
7. (d y)/(d x)=e^3 x+2 y
8. e^x y (d y)/(d x)=e^-y+e^-2 x,
9. y ln x (d x)/(d y)=((y+1)/(x))^2
10. (d y)/(d x)=((2 y+3)/(4 x+5))^2
11. csc y d x+sec ^2 x d y=0
12. sin 3 x d x+2 y cos ^3 3 x d y=0
13. (e^y+1)^2 e^-x d x+(e^x+1)^3 e^-x d y=0
14. x(1+y^2)^1 / 2 d x=y(1+x^2)^1 / 2 d y
15. (d S)/(d r)=k S
16. (d Q)/(d t)=k(Q-70)
17. (d P)/(d t)=P-P^2
18. (d N)/(d t)+N=N t e^t+2
19. (d y)/(d x)=(x y+3 x-y-3)/(x y-2 x+4 y-8)
20. (d y)/(d x)=(x y+2 y-x-2)/(x y-3 y+x-3)
21. (d y)/(d x)=x √(1-y^2)
22. (e^x+e^-x) (d y)/(d x)=y^2

In Problems 23-28 find an explicit solution of the given initial-value problem.
23. (d x)/(d t)=4(x^2+1), x(π / 4)=1
24. (d y)/(d x)=(y^2-1)/(x^2-1), y(2)=2
25. x^2(d y)/(d x)=y-x y, y(-1)=-1
26. (d y)/(d t)+2 y=1, y(0)=(5)/(2)
27. √(1-y^2) d x-√(1-x^2) d y=0, y(0)=(√(3))/(2)
28. (1+x^4) d y+x(1+4 y^2) d x=0, y(1)=0

Separable - Answers

EXERCISES 2.2 (PAGE 52)
1. y=-(1)/(5)cos 5 x+c
3. y=(1)/(3) e^-3 x+c
5. y=c x^4
7. -3 e^-2 y=2 e^3 x+c
9. (1)/(3) x^3ln x-(1)/(9) x^3=(1)/(2) y^2+2 y+ln |y|+c
11. 4 cos y=2 x+sin 2 x+c
13. (e^x+1)^-2+2(e^y+1)^-1=c
15. S=c e^k r
17. P=(c e^t)/(1+c e^t)
19. (y+3)^5 e^x=c(x+4)^5 e^y
21. y=sin((1)/(2) x^2+c)
23. x=tan(4 t-(3)/(4)π)
25. y=(e^-(1+1 / x))/(x)
27. y=(1)/(2) x+(√(5))/(2)√(1-x^2)
29. y=e^∫ i e^-t^2 d t

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