2.1E: Separable Equations (Exercises)
- Page ID
- 103471
\newcommand{\oiint}
Callstack:
at (Template:Custom/Views/ContentHeader), /content/body/div[2]/p[1]/span, line 1, column 1
at (Courses/Cosumnes_River_College/Math_420:_Differential_Equations_(Breitenbach)/02:_First_Order_Equations/2.01:_Separable_Equations/2.1E:_Separable_Equations_(Exercises)), /content/body/p[41]/@class
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
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