A.4.5: Section 4.5 Answers
- Page ID
- 43764
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. \(y'=\frac{2xy}{x^{2}+3y^{2}}\)
2. \(y'=-\frac{y^{2}}{(xy-1)}\)
3. \(y'=-\frac{y(x^{2}+y^{2}-2x^{2}\ln |xy|)}{x(x^{2}+y^{2}-2y^{2}\ln |xy|)}\)
4. \(xy'-y=-\frac{x^{1/2}}{2}\)
5. \(y'+2xy=4xe^{x^{2}}\)
6. \(xy'+y=4x^{3}\)
7. \(y' − y = \cos x − \sin x\)
8. \((1+x^{2})y'-2xy=(1-x)^{2}e^{x}\)
10. \(y'g-yg'=f'g-fg'\)
11. \((x-x_{0})y'=y-y_{0}\)
12. \(y'(y^{2}-x^{2}+1)+2xy=0\)
13. \(2x(y-1)y'-y^{2}+x^{2}+2y=0\)
14.
- \(y=-81+18x,\: (9,81)\quad y=-1+2x,\: (1,1)\)
- \(y = −121 + 22x,\: (11, 121)\quad y = −1 + 2x,\: (1, 1)\)
- \(y = −100 − 20x,\: (−10, 100)\quad y = −4 − 4x,\: (−2, 4)\)
- \(y = −25 − 10x,\: (−5, 25)\quad y = −1 − 2x,\: (−1, 1)\)
15. (e) \(y=\frac{5+3x}{4},\:(-3/5,4/5)\quad y=-\frac{5-4x}{3},\: (4/5,-3/5)\)
17.
- \(y=-\frac{1}{2}(1+x),\: (1,-1);\quad y=\frac{5}{2}+\frac{x}{10},\: (25,5)\)
- \(y=\frac{1}{4}(4+x),\: (4,2)\quad y=-\frac{1}{4}(4+x),\: (4,-2)\)
- \(y = \frac{1}{2}(1 + x),\: (1, 1)\quad y = \frac{7}{2} + \frac{x}{14},\: (49, 7)\)
- \(y = −\frac{1}{2}(1 + x),\: (1, −1)\quad y = −\frac{5}{2} −\frac{x}{10},\: (25, −5)\)
18. \(y=2x^{2}\)
19. \(y=\frac{cx}{\sqrt{|x^{2}=1|}}\)
20. \(y=y_{1}+c(x-x_{1})\)
21. \(y=-\frac{x^{3}}{2}-\frac{x}{2}\)
22. \(y=-x\ln |x|+cx\)
23. \(y=\sqrt{2x+4}\)
24. \(y=\sqrt{x^{2}-3}\)
25. \(y=kx^{2}\)
26. \((y-x)^{3}(y+x)=k\)
27. \(y^{2}=-x+k\)
28. \(y^{2}=-\frac{1}{2}\ln (1+2x^{2})+k\)
29. \(y^{2}=-2x-\ln (x-1)^{2}+k\)
30. \(y=1+\sqrt{\frac{9-x^{2}}{2}};\text{ those with }c>0\)
33. \(\tan ^{-1}\frac{y}{x}-\frac{1}{2}\ln (x^{2}+y^{2})=k\)
34. \(\frac{1}{2}\ln (x^{2}+y^{2})+(\tan\alpha )\tan ^{-1}\frac{y}{x}=k\)