Internal problem ID [10033]
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: 39.
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+\frac {a \left (x -2\right ) y}{x}-\frac {2 a^{2} \left (x -1\right )}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 116
dsolve(y(x)*diff(y(x),x)+a*(x-2)*x^(-1)*y(x)=2*a^2*(x-1)*x^(-1),y(x), singsol=all)
\[ c_{1}+\frac {x \sqrt {\frac {a}{a x +y \relax (x )}}\, \left (a x +y \relax (x )\right ) \left (\int _{}^{\frac {a}{a x +y \relax (x )}}\frac {\sqrt {\textit {\_a} -1}\, {\mathrm e}^{\frac {1}{2 \textit {\_a}}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right )+\sqrt {\frac {\left (1-x \right ) a -y \relax (x )}{a x +y \relax (x )}}\, {\mathrm e}^{\frac {a x +y \relax (x )}{2 a}} y \relax (x )}{\sqrt {\frac {a}{a x +y \relax (x )}}\, \left (a x +y \relax (x )\right ) x} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]+a*(x-2)*x^(-1)*y[x]==2*a^2*(x-1)*x^(-1),y[x],x,IncludeSingularSolutions -> True]
Not solved