9.2: Chapter 2 Answers

Section 2.1

1. \(y=\dfrac{x^3}{3}+\dfrac{x^2}{2}+\dfrac{13}{6}\)

2. \(y=-\dfrac{1}{5}\cos(5x)+\dfrac{11}{5}\)

3. \(y=\dfrac{1}{2}\ln\Big|\dfrac{x-1}{x+1}\Big|\)

4. \(y=e^x+\dfrac{x^2}{2}+9\)

5. \(y=xe^x-e^x+C\)

6. \(y=-\dfrac{1}{2}\cos(t^2)+C\)

7. \(y=\dfrac{1}{2}\ln|t^2+4|+C\)

8. \(y=\dfrac{1}{2}\tan^{-1}\Big(\dfrac{x}{2}\Big)+C\)

9.  \(y=\dfrac{x^3}{6}+2x^2+C_1x+ C_2\)

10.  \(y=\dfrac{1}{4}e^{2x}+\dfrac{x^3}{6}+C_1x+C_2\)

11.  \(y=\dfrac{x^4}{12}+C_1x^2+C_2x+C_3\)

12. \(y=\cos(x)+C_1x^2+C_2x+C_3\)

Section 2.2

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Section 2.3

1. \(y=2\pm\sqrt{2(x^{3}+x^{2}+x+c)}\)

2. \(\ln (|\sin y|)=\cos x+c\)

3. \(y=\frac{c}{x-c}\)

4. \(\frac{(\ln y)^{2}}{2}=-\frac{x^{3}}{3}+c\)

5. \(y^{3}+3\sin y+\ln |y|+\ln (1+x^{2})+\tan ^{-1}x=c\)

6. \((y^2-1)^{-1/2}={1\over x}+c\)

7. \(y=\tan \left(\frac{x^{3}}{3}+c \right)\)

8. \(y=\frac{c}{\sqrt{1+x^{2}}}\)

9. \({y-2\over y-1}=ce^{(x^2-2x)/2}\)

10. \(y=1+(3x^{2}+9x+c)^{1/3}\)

11. \(y=2+\sqrt{\frac{2}{3}x^{3}+3x^{2}+4x-\frac{11}{3}}\) 

12. \(xy=e^{-(1+1/x)}\)

13. \(y^{3}+2y^{2}+x^{2}+\sin x=3\)

14. \((y+1)(y-1)^{-3}(y-2)^{2}=-256(x+1)^{-6}\)

15. \(y=-1+3e^{-x^{2}}\)

16. \({y^2\over y^2+1}={1\over 2}e^{2x^2}\)

17. \({y-2\over y-1}={1\over 2}e^{-x^2};\quad (-\infty,\infty)\)

18. \(y=\frac{-1+\sqrt{4x^{2}-15}}{2};\quad ({\sqrt {15}\over 2},\infty)\)

19. \(y=\frac{2}{1+e^{-2x}}; \quad (-\infty ,\infty )\)

20. \(y=-\sqrt{25-x^{2}};\quad (-5,5)\)

21. \(y≡2;\quad (-\infty ,\infty )\)

22. \(y=3\left(\frac{x+1}{2x-4} \right)^{1/3};\quad (-\infty ,2)\)

23. \(y=\frac{x+c}{1-cx}\)

24. \(y=-x\cos c+\sqrt{1-x^{2}}\sin c\)

25. \(y=-x+3\pi /2\)

Section 2.4

1. \(y=Ce^{-x^3}\)

2. \(x={1\over 4}e^{3t}+ce^{-t}\)

3. \(y={1\over 2}x^2-{1\over 2}+ce^{-x^2}\)

4. \(y=\sin x+c\cos x\)

5. \(y=x^{-1}\ln x+cx^{-1}\)

6. \((\sec \theta+\tan \theta)r=\theta-\cos \theta +c\)

7. \(y=\frac{2}{x}+\frac{c}{x}e^{x}\)

8. \(y=e^{-x^{2}}\left(\frac{x^{2}}{2}+c \right)\)

9. \(y=-\frac{e^{-x}+c}{1+x^{2}}\)

10. \(\frac{7\ln |x|}{x}+\frac{3}{2}x+\frac{c}{x}\)

11. \(y=(x-1)^{-4}(\ln |x-1|-\cos x+c)\)

12. \(y=e^{-x^{2}}\left(\frac{x^{3}}{4}+\frac{c}{x} \right)\)

13. \(y=\frac{2\ln |x|}{x^{2}}+\frac{1}{2}+\frac{c}{x^{2}}\)

14. \(y=(x+c)\cos x\)

15. \(y=\frac{c-\cos x}{(1+x)^{2}}\)

16. \(y=-\frac{1}{2}\frac{(x-2)^{3}}{(x-1)}+c\frac{(x-2)^{5}}{(x-1)}\)

17. \(y=(x+c)e^{-\sin ^{2}x}\)

18. \(y=\frac{e^{x}}{x^{2}}-\frac{e^{x}}{x^{3}}+\frac{c}{x^{2}}\)

19. \(y=\frac{e^{-(x-1)}}{x}\)

20. \(y=\frac{e}{x\ln x}\)

21. \(y=\frac{\pi }{x\sin x}\)

22. \(y=2(1+x^{2})\)

23. \(y=\frac{e^{3x}-e^{-7x}}{10}\)

24. \(y=\frac{2x+1}{(1+x^{2})^{2}}\)

25. \(y=\frac{1}{x^{3}}\ln \left(\frac{1+x^{2}}{2} \right)\)

26. \(y=\frac{1}{2}(\sin x +\csc x)\)

27. \(y=\frac{2\ln |x|}{x}+\frac{x}{2}-\frac{1}{2x}\)

28. \(y=(x-1)^{-3}[\ln (1-x)-\cos x]\)

29. \(y=2x^{2}+\frac{1}{x^{2}}\)

30. \(y=x^{2}(1-\ln x)\)

31. \(y=\frac{1}{2}+\frac{5}{2}e^{-x^{2}}\)

32. \(y=\frac{\ln |x-1|+\tan x+1}{(x-1)^{3}}\)

33. \(y=\frac{\ln |x|+x^{2}+1}{(x+2)^{4}}\)

Section 2.5

1. \(2x^{3}y^{2}=c\)

2. \(3y\sin x+2x^{2}e^{x}+3y=c\)

3. Not exact

4. \(x^{2}-2xy^{2}+4y^{3}=c\)

5. \(x+y=c\)

6. Not exact

7. \(y\sin^2 x-xy+2e^{xy^2}=c\)

8. \(2y^{2}\cos x+3xy^{3}-x^{2}=c\)

9. Not exact

10. \(x^{3}+x^{2}y+4xy^{2}+9y^{2}=c\)

11. Not exact

12. \(\ln |xy|+x^{2}+y^{2}=c\)

13. Not exact

14. \(x^{2}+y^{2}=c\)

15. \(x^{2}y^{2}e^{x}+2y+3x^{2}=c\)

16. \(x^{3}e^{x^{2}+y}-4y^{3}+2x^{2}=c\)

17. \(x^{4}e^{xy}+3xy=c\)

18. \(x^{3}\cos xy+4y^{2}+2x^{2}=c\)

19. \(y=\frac{x+\sqrt{2x^{2}+3x-1}}{x^{2}}\)

20. \(y=\sin x-\sqrt{1-\frac{\tan x}{2}}\)

21. \(y=\left(\frac{e^{x}-1}{e^{x}+1} \right)^{1/3}\)

22. \(y=1+2\tan x\)

23. \(y=\frac{x^{2}-x+6}{(x+2)(x-3)}\)

Section 2.6

1. \(y=\frac{1}{1-ce^{x}}\)

2. \(y=x^{2/7}(c-\ln |x|)^{1/7}\)

3. \(y=e^{2/x}(c-1/x)^{2}\)

4. \(y=\pm\frac{\sqrt{2x+c}}{1+x^{2}}\)

5. \(y=\pm (1-x^{2}+ce^{-x^{2}})^{-1/2}\)

6. \(y=\left[\frac{x}{3(1-x)+ce^{-x}} \right] ^{1/3}\)

7. \(y=\frac{2\sqrt{2}}{\sqrt{1-4x}}\)

8. \(y=\left[-1+\frac{3}{2}e^{-(x^{2}-1)/4} \right]^{-2}\)

9. \(y=\frac{1}{x(11-3x)^{1/3}}\)

10. \(y=(2e^{x}-1)^{2}\)

11. \(y=(2e^{12x}-1-12x)^{1/3}\)

12. \(y=\left[\frac{5x}{2(1+4x^{5})} \right]^{1/2}\)

13. \(y=(4e^{x/2}-x-2)^{2}\)

14. \(y=x(\ln |x|+c)\)

15. \(\ln |{y\over x}|-\ln |{y\over x}+1|=\ln |x|+c; \quad y=-x\)

16. \(y=\pm x(4\ln |x|+c)^{1/4}\)

17. \(y=x\sin ^{-1}(\ln |x|+c)\)

18. \(y=x\tan (\ln |x|+c)\)

19. \(y=\pm x\sqrt{cx^{2}-1}\)

20. \(y=-\frac{2x}{2\ln |x|+1}\)

21. \(y=x(3\ln x+27)^{1/3}\)

22. \(y=\frac{1}{x}\left(\frac{9-x^{4}}{2} \right)^{1/2}\)

23. \(y=-x\)

24. \(y=-\frac{x(4x-3)}{(2x-3)}\)

25. \(y=x\sqrt{4x^{6}-1}\)

26. \(y=-x-1+\tan (x+c)\)

27. \(2y-2x+\sin 2(x+y)=c\)

28. \(4y-8x+12=(x+c)^2\)

29. \((x+y)^2=2x+c\)

30. \(\tan (x+y)-\sec (x+y)=x+c\)

31. \(-e^{-y+x-5}=x+c\)

32. \(-\cot (x+y)+\csc (x+y)=x+\sqrt{2}-1\)

33. \(4\ln |15x+10y+6|=10x-10y+4\ln 19\)

Section 2.7

1. \(y_{1}= 1.450000000,\: y_{2} = 2.085625000,\: y_{3} = 3.079099746\)

2. \(y_{1} = 1.200000000,\: y_{2} = 1.440415946,\: y_{3} = 1.729880994\)

3. \(y_{1} = 1.900000000,\: y_{2} = 1.781375000,\: y_{3} = 1.646612970\)

4. \(y_{1} = 2.962500000,\: y_{2} = 2.922635828,\: y_{3} = 2.880205639\)

5. \(y_{1} = 2.513274123,\: y_{2} = 1.814517822,\: y_{3} = 1.216364496\)

6.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(1.0\)	\(48.298147362\)	\(51.492825643\)	\(53.076673685\)	\(54.647937102\)

7.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(2.0\)	\(1.390242009\)	\(1.370996758\)	\(1.361921132\)	\(1.353193719\)

8.

\(x\)	\(h=0.05\)	\(h=0.025\)	\(h=0.0125\)	Exact
\(1.50\)	\(7.886170437\)	\(8.852463793\)	\(9.548039907\)	\(10.500000000\)