Internal problem ID [10009]
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: 15.
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 (\left (m -1\right ) x +1\right ) y}{x}-\frac {a^{2} \left (m x +1\right ) \left (x -1\right )}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 521
dsolve(y(x)*diff(y(x),x)-a*((m-1)*x+1)*1/x*y(x)=a^2*1/x*(m*x+1)*(x-1),y(x), singsol=all)
\[ c_{1}-\frac {9 \left (3 m x -m +1\right ) m \left (\frac {\left (a m x +a -y \relax (x )\right ) m}{a m -y \relax (x ) m +2 a -2 y \relax (x )}\right )^{\frac {m}{m +1}} \left (\frac {x a \,m^{2}}{a m -y \relax (x ) m -a +y \relax (x )}\right )^{-\frac {m}{m +1}} \left (\frac {\left (a x -a +y \relax (x )\right ) m^{2}}{2 a m -2 y \relax (x ) m +a -y \relax (x )}\right )^{\frac {1}{m +1}} \left (\frac {x a \,m^{2}}{a m -y \relax (x ) m -a +y \relax (x )}\right )^{-\frac {1}{m +1}}}{2 m^{3}+3 m^{2}-3 m -2}-\left (\int _{}^{-\frac {9 m \left (3 a m x +y \relax (x ) m -a m -y \relax (x )+a \right )}{2 y \relax (x ) m^{3}-2 a \,m^{3}+3 y \relax (x ) m^{2}-3 a \,m^{2}-3 y \relax (x ) m +3 a m -2 y \relax (x )+2 a}}\frac {\textit {\_a} \left (\textit {\_a} \,m^{2}+\textit {\_a} m -2 \textit {\_a} -9 m \right )^{\frac {m^{2}}{\left (m +1\right ) \left (m^{2}+m -2\right )}} \left (\textit {\_a} \,m^{2}+\textit {\_a} m -2 \textit {\_a} -9 m \right )^{\frac {m}{\left (m +1\right ) \left (m^{2}+m -2\right )}} \left (\textit {\_a} \,m^{2}+\textit {\_a} m -2 \textit {\_a} -9 m \right )^{-\frac {2}{\left (m +1\right ) \left (m^{2}+m -2\right )}} \left (2 \textit {\_a} \,m^{2}-\textit {\_a} m -\textit {\_a} +9 m \right )^{\frac {2 m^{3}}{\left (m +1\right ) \left (2 m^{2}-m -1\right )}} \left (2 \textit {\_a} \,m^{2}-\textit {\_a} m -\textit {\_a} +9 m \right )^{-\frac {m^{2}}{\left (m +1\right ) \left (2 m^{2}-m -1\right )}} \left (2 \textit {\_a} \,m^{2}-\textit {\_a} m -\textit {\_a} +9 m \right )^{-\frac {m}{\left (m +1\right ) \left (2 m^{2}-m -1\right )}}}{\left (2 \textit {\_a} \,m^{2}+5 \textit {\_a} m +2 \textit {\_a} +9 m \right ) \left (-\frac {\left (m -1\right )^{2} \left (1+2 m \right )^{2} \left (m +2\right )^{2} \textit {\_a}^{3}}{729 m^{3}}+\frac {\left (m^{2}+m +1\right ) \textit {\_a}}{3 m}+1\right )}d \textit {\_a} \right ) = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*((m-1)*x+1)*1/x*y[x]==a^2*1/x*(m*x+1)*(x-1),y[x],x,IncludeSingularSolutions -> True]
Not solved