Internal problem ID [1078]
Internal file name [OUTPUT/1079_Sunday_June_05_2022_02_02_07_AM_94834817/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Exact equations. Integrating factors. Section 2.6 Page
91
Problem number: 19.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {3 y^{3} x^{2}-y^{2}+y+\left (-y x +2 x \right ) y^{\prime }=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 y^{3} x^{2}-y^{2}+y+\left (-y x +2 x \right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-3 y^{3} x^{2}+y^{2}-y}{-y x +2 x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 59
dsolve((3*x^2*y(x)^3-y(x)^2+y(x))+(-x*y(x)+2*x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {4}{\sqrt {x}\, \sqrt {\frac {c_{1} x +48 x^{2}+4}{x}}+2} \\ y \left (x \right ) &= -\frac {4}{\sqrt {x}\, \sqrt {\frac {c_{1} x +48 x^{2}+4}{x}}-2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.776 (sec). Leaf size: 80
DSolve[(3*x^2*y[x]^3-y[x]^2+y[x])+(-x*y[x]+2*x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2}{1+\sqrt {-\frac {1}{x^2}} x \sqrt {-12 x^2-4 c_1 x-1}} \\ y(x)\to \frac {2 x}{x+\frac {\sqrt {-12 x^2-4 c_1 x-1}}{\sqrt {-\frac {1}{x^2}}}} \\ y(x)\to 0 \\ \end{align*}