Internal problem ID [10661]
Internal file name [OUTPUT/9609_Monday_June_06_2022_03_13_33_PM_8713158/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. Form \(y y'-y=f(x)\). subsection 1.3.1-2.
Solvable equations and their solutions
Problem number: 13.
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 B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=\frac {\left (1+2 m \right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y y^{\prime } x^{3} m^{2}-4 y x^{3} m^{2}-4 A \,x^{2} m^{2}-2 m \,x^{4}+4 m^{2} A^{2}-x^{4}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {4 y x^{3} m^{2}+4 A \,x^{2} m^{2}+2 m \,x^{4}-4 m^{2} A^{2}+x^{4}}{4 y x^{3} m^{2}} \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 166
dsolve(y(x)*diff(y(x),x)-y(x)=(2*m+1)/(4*m^2)*x+A*1/x-A^2*1/(x^3),y(x), singsol=all)
\[ \frac {2^{-\frac {m}{1+m}} y \left (x \right ) \left (\frac {-2 y \left (x \right ) m x -2 A m -x^{2}}{2 y \left (x \right ) x +2 A}\right )^{\frac {1}{1+m}} \left (y \left (x \right ) x +A \right ) \left (\frac {\left (-1-2 m \right ) x^{2}+2 y \left (x \right ) m x +2 A m}{y \left (x \right ) x +A}\right )^{\frac {1+2 m}{1+m}}-x \left (A \left (\int _{}^{-\frac {x^{2}}{2 y \left (x \right ) x +2 A}}\frac {\left (-m +\textit {\_a} \right )^{\frac {1}{1+m}} \left (\left (2 \textit {\_a} +1\right ) m +\textit {\_a} \right )^{\frac {1+2 m}{1+m}}}{\textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} \right )}{x} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==(2*m+1)/(4*m^2)*x+A*1/x-A^2*1/(x^3),y[x],x,IncludeSingularSolutions -> True]
Not solved