Internal problem ID [10653]
Internal file name [OUTPUT/9601_Monday_June_06_2022_03_13_16_PM_16781380/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: 5.
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.
\[ \boxed {y y^{\prime }-y=A x +\frac {B}{x}-\frac {B^{2}}{x^{3}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & A \,x^{4}-y y^{\prime } x^{3}+y x^{3}+B \,x^{2}-B^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-A \,x^{4}-y x^{3}-B \,x^{2}+B^{2}}{y x^{3}} \end {array} \]
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: 179
dsolve(y(x)*diff(y(x),x)-y(x)=A*x+B/x-B^2*x^(-3),y(x), singsol=all)
\[ \frac {\left (-y \left (x \right ) x^{2} B -B^{2} x \right ) \left (\int _{}^{-\frac {x^{2}}{2 y \left (x \right ) x +2 B}}\frac {{\mathrm e}^{\frac {2 \,\operatorname {arctanh}\left (\frac {4 A \textit {\_a} -1}{\sqrt {4 A +1}}\right )}{\sqrt {4 A +1}}} \left (4 A \,\textit {\_a}^{2}-2 \textit {\_a} -1\right )}{\textit {\_a}^{2}}d \textit {\_a} \right )+2 y \left (x \right ) \left (-y \left (x \right )^{2} x^{2}+\left (x^{3}-2 B x \right ) y \left (x \right )+A \,x^{4}+B \,x^{2}-B^{2}\right ) {\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 A \,x^{2}+y \left (x \right ) x +B}{\sqrt {4 A +1}\, \left (y \left (x \right ) x +B \right )}\right )}{\sqrt {4 A +1}}}+x c_{1} \left (y \left (x \right ) x +B \right )}{x \left (y \left (x \right ) x +B \right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==A*x+B/x-B^2*x^(-3),y[x],x,IncludeSingularSolutions -> True]
Not solved