# 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.

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

This page titled A.2.1: Section 2.1 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.