2.4E: Solving Differential Equations by Substitution (Exercises)
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- 103475
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Bernoulli Equations
In Exercises 1-8 solve the given equation.
1. \(y'+y=y^2\)
2. \( {7x{dy\over dx}-2y=-{x^2 \over y^6}}\)
3. \(x^2y'+2y=2e^{1/x}y^{1/2}\)
4. \({dy\over dx}=y(xy^3-1)\)
5. \( {(1+x^2)y'+2xy ={1 \over (1+x^2)y}}\)
6. \(t^2{dy\over dt}+y^2=ty\)
7. \(y'-xy=x^3y^3\)
8. \( {{dy\over dx}-{1+x\over 3x}y=y^4}\)
In Exercises 9-16 solve the initial value problem.
9. \(y'-2y=xy^3,\quad y(0)=2\sqrt2\)
10. \(y'-xy=xy^{3/2},\quad y(1)=4\)
11. \(xy'+y=x^4y^4,\quad y(1)=1/2\)
12. \(y'-2y=2y^{1/2},\quad y(0)=1\)
13. \(x^2{dy\over dx}-2xy=3y^4, \quad y(1)={1\over 2}\)
14. \( {y'-4y={48x\over y^2},\quad y(0)=1}\)
15. \(x^2y'+2xy=y^3,\quad y(1)=1/\sqrt2\)
16. \(y'-y=xy^{1/2},\quad y(0)=4\)
Homogeneous Equations
In Exercises 17-24 solve the equation.
17. \(y'= {y+x\over x}\)
18. \(y'= {y^2+2xy \over x^2}\)
19. \(xy^3y'=y^4+x^4\)
20. \(y'= {y\over x}+\sec{y\over x}\)
21. \(x^2y'=xy+x^2+y^2\)
22. \(-ydx+(x+{\sqrt {xy}})dy=0\)
23. \(xyy'=x^2+2y^2\)
24. \(y'= {2y^2+x^2e^{-(y/x)^2}\over 2xy}\)
In Exercises 25-32 solve the initial value problem.
25. \(y'= {xy+y^2\over x^2}, \quad y(-1)=2\)
26. \(xy^2{dy\over dx}=y^3-x^3, \quad y(1)=2\)
27. \(y'= {x^3+y^3\over xy^2}, \quad y(1)=3\)
28. \(xyy'+x^2+y^2=0, \quad y(1)=2\)
29. \((x+ye^{y\over x})dx-xe^{y\over x}dy=0, \quad y(1)=0\)
30. \(y'= {y^2-3xy-5x^2 \over x^2}, \quad y(1)=-1\)
31. \(x^2y'=2x^2+y^2+4xy, \quad y(1)=1\)
32. \(xyy'=3x^2+4y^2, \quad y(1)=\sqrt{3}\)
In Exercises 33-39 solve the given equation.
33. \(dy= {x+y \over x-y}dx\)
34. \((y'x-y)(\ln |y|-\ln |x|)=x\)
35. \(y'= {y^3+2xy^2+x^2y+x^3\over x(y+x)^2}\)
36. \((2x+y)dy=(x+2y)dx\)
37. \({dy\over dx}= {y \over y-2x}\)
38. \(y'= {xy^2+2y^3\over x^3+x^2y+xy^2}\)
39. \((x^3+3xy^2){dy\over dx}= x^3+x^2y+3y^3\)
The Substitution u=Ax+By+C
In Exercises 40-45 solve the given equation.
40. \({dy\over dx}=(x+y+1)^2\)
41. \({dy\over dx}=\tan^2 (x+y)\)
42. \({dy\over dx}=2+\sqrt {y-2x+3}\)
43. \({dy\over dx}={1-x-y\over x+y}\)
44. \({dy\over dx}=\sin (x+y)\)
45. \({dy\over dx}=1+e^{y-x+5}\)
In Exercises 46-47 solve the initial value problem.
46. \({dy\over dx}=\cos (x+y), \quad y(0)=\pi/4\)
47. \({dy\over dx}={3x+2y\over 3x+2y+2}, \quad y(-1)=-1\)