A.2.1: Section 2.1 Answers

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1. \(y=e^{-ax}\)

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

3. \(y=ce^{-(\ln x)^{2}/2 }\)

4. \(y=\frac{c}{x^{3}}\)

5. \(y=ce^{1/x}\)

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

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

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

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

10. \(y=3x^{-k}\)

11. \(y=2(\cos kx)^{1/k}\)

12. \(y=\frac{1}{3}+ce^{-3x}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

31. \(y=2x^{2}+\frac{1}{x^{2}}\quad (0,\infty )\)

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

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

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

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

36. \(y=(x^{2}-1)\left(\frac{1}{2}\ln |x^{2}-1|-4 \right)\)

37. \(y=-(x^{2}-5)(7+\ln |x^{2}-5|)\)

38. \(y=e^{-x^{2}}\left(3+\int _{0}^{x} t^{2}e^{t^{2}}dt \right)\)

39. \(y=\frac{1}{x}\left(2+\int_{1}^{x}\frac{\sin t}{t}dt \right)\)

40. \(y=e^{-x}\int_{1}^{x}\frac{\tan t}{t}dt\)

41. \(y=\frac{1}{1+x^{2}}\left(1+\int_{0}^{x}\frac{e^{t}}{1+t^{2}}dt \right)\)

42. \(y=\frac{1}{x}\left( 2e^{-(x-1)}+e^{-x}\int_{1}^{x}e^{t}e^{t^{2}}dt \right)\)

43. \(G=\frac{r}{\lambda }+\left( G_{0}-\frac{r}{\lambda} \right)e^{-\lambda t}\lim _{t\to\infty}G(t)=\frac{r}{\lambda}\)

45. \(y=y_{0}e^{-a(x-x_{0})}+e^{-ax}\int_{x_{0}}^{x}e^{at}f(t)dt\)

48.

\(y=\tan ^{-1}\left(\frac{1}{3}+ce^{3x} \right) \)
\(y=\pm\left[\ln\left(\frac{1}{x}+\frac{c}{x^{2}} \right) \right]^{1/2}\)
\(y=\text{exp}\left(x^{2}+\frac{c}{x^{2}} \right)\)
\(y=-1+\frac{x}{c+3\ln |x|}\)