Internal problem ID [9911]
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.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-y-x A -\frac {B}{x}+\frac {B^{2}}{x^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 171
dsolve(y(x)*diff(y(x),x)-y(x)=A*x+B/x-B^2*x^(-3),y(x), singsol=all)
\[ c_{1}+\frac {\left (-y \relax (x ) B \,x^{2}-B^{2} x \right ) \left (\int _{}^{-\frac {x^{2}}{2 x y \relax (x )+2 B}}\frac {{\mathrm e}^{\frac {2 \arctanh \left (\frac {4 A \textit {\_a} -1}{\sqrt {4 A +1}}\right )}{\sqrt {4 A +1}}} \left (4 A \,\textit {\_a}^{2}-2 \textit {\_a} -1\right )}{\textit {\_a}^{2}}d \textit {\_a} \right )+2 \,{\mathrm e}^{-\frac {2 \arctanh \left (\frac {2 A \,x^{2}+x y \relax (x )+B}{\sqrt {4 A +1}\, \left (x y \relax (x )+B \right )}\right )}{\sqrt {4 A +1}}} \left (A \,x^{4}+x^{3} y \relax (x )+\left (-y \relax (x )^{2}+B \right ) x^{2}-2 B x y \relax (x )-B^{2}\right ) y \relax (x )}{x \left (x y \relax (x )+B \right )} = 0 \]
✗ 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