Internal problem ID [10802]
Internal file name [OUTPUT/9750_Wednesday_June_08_2022_05_54_02_PM_41902957/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 66.
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 A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-a \left (\frac {n +2}{n}+b \,x^{n}\right ) y=-\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y x^{n} a b \,n^{2}-x^{n} a^{2} b n x -y y^{\prime } n^{2}+y a \,n^{2}-a^{2} n x +2 y a n -a^{2} x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y x^{n} a b \,n^{2}+x^{n} a^{2} b n x -y a \,n^{2}+a^{2} n x -2 y a n +a^{2} x}{y n^{2}} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 192
dsolve(y(x)*diff(y(x),x)-a*((n+2)/n+b*x^n)*y(x)=-a^2/n*x*((n+1)/n+b*x^n),y(x), singsol=all)
\[ -n \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, \left (\int _{}^{\frac {2 \arctan \left (\frac {2 x^{n +1} a b n +\left (n +1\right ) \left (a x -y \left (x \right ) n \right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n \left (a x -y \left (x \right ) n \right )}\right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}}\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right )+\left (-2 b n \,x^{n}-n -1\right ) {\mathrm e}^{-\frac {2 \arctan \left (\frac {2 x^{n +1} a b n +\left (n +1\right ) \left (a x -y \left (x \right ) n \right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n \left (a x -y \left (x \right ) n \right )}\right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}}+c_{1} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*((n+2)/n+b*x^n)*y[x]==-a^2/n*x*((n+1)/n+b*x^n),y[x],x,IncludeSingularSolutions -> True]
Not solved