A.2.4: Section 2.4 Answers
- Page ID
- 43754
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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. \(P=\frac{P_{0}e^{at}}{1+aP_{0}\int_{0}^{t}\alpha (\tau )e^{a\tau }d\tau };\quad\lim_{t\to\infty }P(t)=\left\{\begin{array}{cc}{\infty }&{\text{if }L=0,}\\[4pt]{0}&{\text{if }L=\infty ,}\\[4pt]{1/aL}&{\text{if }0<L<\infty }\end{array} \right.\)
15. \(y=x(\ln |x|+c)\)
16. \(y=\frac{cx^{2}}{1-cx}\quad y=-x\)
17. \(y=\pm x(4\ln |x|+c)^{1/4}\)
18. \(y=x\sin ^{-1}(\ln |x|+c)\)
19. \(y=x\tan (\ln |x|+c)\)
20. \(y=\pm x\sqrt{cx^{2}-1}\)
21. \(y=\pm x\ln (\ln |x|+c)\)
22. \(y=-\frac{2x}{2\ln |x|+1}\)
23. \(y=x(3\ln x+27)^{1/3}\)
24. \(y=\frac{1}{x}\left(\frac{9-x^{4}}{2} \right)^{1/2}\)
25. \(y=-x\)
26. \(y=-\frac{x(4x-3)}{(2x-3)}\)
27. \(y=x\sqrt{4x^{6}-1}\)
28. \(\tan ^{-1}\frac{y}{x}-\frac{1}{2}\ln (x^{2}+y^{2})=c\)
29. \((x+y)\ln |x|+y(1-\ln |y|)+cx=0\)
30. \((y+x)^{3}=3x^{3}(\ln |x|+c)\)
31. \((y+x)=c(y-x)^{3};\quad y=x;\quad y=-x\)
32. \(y^{2}(y-3x)=c;\quad y≡0;\quad y=3x\)
33. \((x-y)^{3}(x+y)=cy^{2}x^{4};\quad y=0;\quad y=x;\quad y=-x\)
34. \(\frac{y}{x}+\frac{y^{3}}{x^{3}}=\ln |x|+c\)
40. Choose \(X_{0}\) and \(Y_{0}\) so that
\[aX_{0}+bY_{0}=\alpha\nonumber \] \[cX_{0}+dY_{0}=\beta\nonumber \]
41. \((y+2x+1)^{4})2y-6x-3)=c;\quad y=3x+3/2;\quad y=-2x-1\)
42. \((y+x-1)(y-x-5)^{3}=c;\quad y=x+5;\quad y=-x+1\)
43. \(\ln |y-x-6|-\frac{2(x+2)}{y-x-6}=c;\quad y=x+6\)
44. \((y_{1}=x^{1/3}y=x^{1/3}(\ln |x|+c)^{1/3}\)
45. \(y_{1}=x^{3};\quad y=\pm x^{3}\sqrt{cx^{6}-1}\)
46. \(y_{1}=x^{2};\quad y=\frac{x^{2}(1+cx^{4})}{1-cx^{4}} y=-x^{2}\)
47. \(y_{1}=e^{x};\quad y=-\frac{e^{x}(1-2ce^{x}}{1-ce^{x}};\quad y=-2e^{x}\)
48. \(y_{1}=\tan x; y=\tan x\tan (\ln |\tan x|+c)\)
49. \(y_{1}=\ln x;\quad y=\frac{2\ln x(1+c(\ln x)^{4})}{1-c(\ln x)^{4}};\quad y=-2\ln x\)
50. \(y_{1}=x^{1/2};\quad y=x^{1/2}(-2\pm\sqrt{\ln |x|+c})\)
51. \(y_{1}=e^{x^{2}};\quad y=e^{x^{2}}(-1\pm\sqrt{2x^{2}+c})\)
52. \(y=\frac{-3+\sqrt{1+60x}}{2x}\)
53. \(y=\frac{-5+\sqrt{1+48x}}{2x^{2}}\)
56. \(y=1+\frac{1}{x+1+ce^{x}}\)
57. \(y=e^{x}-\frac{1}{1+ce^{-x}}\)
58. \(y=1-\frac{1}{x(1-cx)}\)
59. \(y=x-\frac{2x}{x^{2}+c}\)