2.1E: Separable Equations (Exercises)
- Page ID
- 103471
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Exercises 1-12, find all solutions.
1. \( {y'={3x^2+2x+1\over y-2}}\)
2. \((\sin x)(\sin y)+(\cos y){dy \over dx}=0\)
3. \({dy \over dx}=e^{2x+5y}\)
4. \(xy'+y^2+y=0\)
5. \({dy \over dx}=y-y^2\)
6. \(y' \ln y+x^2y= 0\)
7. \( {(3y^3+3y \cos y+1)dy+{(2x+1)y\over 1+x^2}dx=0}\)
8. \(x^2yy'=(y^2-1)^{3/2}\)
9. \( {{dy \over dx}=x^2(1+y^2)}\)
10. \((1+x^2)dy+xy dx=0\)
11. \(y'=(x-1)(y-1)(y-2)\)
12. \((y-1)^2y'=2x+3\)
In Exercises 13-19 solve the initial value problem.
13. \( {{dy \over dx}={x^2+3x+2\over y-2}, \quad y(1)=4}\)
14. \(y'+x(y^2+y)=0, \quad y(2)=1\)
15. \(x^2{dy \over dx}=y-xy, \quad y(-1)=-1\)
16. \((3y^2+4y)dy+(2x+\cos x)dx=0, \quad y(0)=1\)
17. \( {y'+{(y+1)(y-1)(y-2)\over x+1}=0, \quad y(1)=0}\)
18. \(y'+2x(y+1)=0, \quad y(0)=2\)
19. \({dy \over dx}=2xy(1+y^2),\quad y(0)=1\)
In Exercises 20-27 solve the initial value problem and give the interval of validity of the solution.
20. \((x^2+2)dy+ 4x(y^2+2y+1)dx=0, \quad y(1)=-1\)
21. \(y'=-2x(y^2-3y+2), \quad y(0)=3\)
22. \( {y'={2x\over 1+2y}, \quad y(2)=0}\)
23. \({dy \over dx}=2y-y^2, \quad y(0)=1\)
24. \(x+yy'=0, \quad y(3) =-4\)
25. \(y'+x^2(y+1)(y-2)^2=0, \quad y(4)=2\)
26. \((x+1)(x-2)dy+ydx=0, \quad y(1)=-3\)
27. \({dy \over dx}={-x \over y}, \quad y(4)=-3\)
28. Solve \( {y'={(1+y^2) \over (1+x^2)}}\) explicitly.
29. Solve \( {{dy \over dx}\sqrt{1-x^2}+\sqrt{1-y^2}=0}\) explicitly.
30. Solve \( {y'={\cos x\over \sin y},\quad y (\pi)={\pi\over2}}\) explicitly.
31. Solve \({dy \over dx}=y^2-y\) explicitly for the initial condition and give the interval of validity of the solution.
a. \(y(0)=2\)
b. \(y(0)=0\)
32. Solve \({dy \over dx}=(y-1)^2\) for the initial condition
a. \(y(0)=0\)
b. \(y(0)=1\)
33. From Theorem 1.2.1, the initial value problem \[y'=3x(y-1)^{1/3}, \quad y(0)=9\nonumber\] has a unique solution on an open interval that contains \(x_0=0\). Find the solution and determine the largest open interval on which it is unique.
34. From Theorem 1.2.1, the initial value problem \[y'=3x(y-1)^{1/3}, \quad y(3)=-7 \nonumber \] has a unique solution on some open interval that contains \(x_0=3\). Find the solution and determine the largest open interval on which it is unique