61.22.5 problem 5

Internal problem ID [12330]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 07:54:38 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y-y&=A x +\frac {B}{x}-\frac {B^{2}}{x^{3}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (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)
 

\[ \frac {\left (-y B \,x^{2}-B^{2} x \right ) \left (\int _{}^{-\frac {x^{2}}{2 x y+2 B}}\frac {{\mathrm e}^{\frac {2 \,\operatorname {arctanh}\left (\frac {4 A \textit {\_a} -1}{\sqrt {4 A +1}}\right )}{\sqrt {4 A +1}}} \left (4 A \,\textit {\_a}^{2}-2 \textit {\_a} -1\right )}{\textit {\_a}^{2}}d \textit {\_a} \right )+2 \left (-x^{2} y^{2}+\left (x^{3}-2 B x \right ) y+A \,x^{4}+B \,x^{2}-B^{2}\right ) y \,{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 A \,x^{2}+x y+B}{\sqrt {4 A +1}\, \left (x y+B \right )}\right )}{\sqrt {4 A +1}}}+x c_{1} \left (x y+B \right )}{x \left (x y+B \right )} = 0 \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[y[x]*D[y[x],x]-y[x]==A*x+B/x-B^2*x^(-3),y[x],x,IncludeSingularSolutions -> True]
 

Not solved