6: Section 2.3 Answers
- Page ID
- 103621
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1. \(y=e^{-ax}\)
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. \(x=2y^6+cy^4\)
12. \(y=(x-1)^{-4}(\ln |x-1|-\cos x+c)\)
13. \(y=e^{-x^{2}}\left(\frac{x^{3}}{4}+\frac{c}{x} \right)\)
14. \(y=\frac{2\ln |x|}{x^{2}}+\frac{1}{2}+\frac{c}{x^{2}}\)
15. \(p=(t+c)\cos t\)
16. \(y=\frac{c-\cos x}{(1+x)^{2}}\)
17. \(y=-\frac{1}{2}\frac{(x-2)^{3}}{(x-1)}+c\frac{(x-2)^{5}}{(x-1)}\)
18. \(x={1\over 2y^2}e^y+{c\over y^2}e^{-y}\)
19. \(y=(x+c)e^{-\sin ^{2}x}\)
20. \(y=\frac{e^{x}}{x^{2}}-\frac{e^{x}}{x^{3}}+\frac{c}{x^{2}}\)
21. \(y=\frac{e^{-(x-1)}}{x}\)
22. \(y=\frac{e}{x\ln x}\)
23. \(y=\frac{\pi }{x\sin x}\)
24. \(y=2(1+x^{2})\)
25. \(x=3t^{-k}\)
26. \(y=2(\cos kx)^{1/k}\)
27. \(y=\frac{e^{3x}-e^{-7x}}{10}\)
28. \(y=\frac{2x+1}{(1+x^{2})^{2}}\)
29. \(y=\frac{1}{x^{3}}\ln \left(\frac{1+x^{2}}{2} \right)\)
30. \(y=\frac{1}{2}(\sin x +\csc x)\)
31. \(y=\frac{2\ln |x|}{x}+\frac{x}{2}-\frac{1}{2x}\)
32. \(y=(x-1)^{-3}[\ln (1-x)-\cos x]\)
33. \(x=2y^{2}+\frac{1}{y^{2}}\)
34. \(y=x^{2}(1-\ln x)\)
35. \(y=\frac{1}{2}+\frac{5}{2}e^{-x^{2}}\)
36. \(y=\frac{\ln |x-1|+\tan x+1}{(x-1)^{3}}\)
37. \(y=\frac{\ln |x|+x^{2}+1}{(x+2)^{4}}\)
38. \(x=(t^{2}-1)\left(\frac{1}{2}\ln |t^{2}-1|-4 \right)\)
39. \(y=-(x^{2}-5)(7+\ln |x^{2}-5|)\)
40. \((x+1)y=x\ln x-x+21\)
41. \(G=\frac{r}{\lambda }+\left( G_{0}-\frac{r}{\lambda} \right)e^{-\lambda t}\lim _{t\to\infty}G(t)=\frac{r}{\lambda}\)