61.24.15 problem 15

Internal problem ID [12428]
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.
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 07:58:37 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 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} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 279

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)
 

\[ -\frac {27 \left (m -1\right ) \left (-54 x \left (\frac {1}{2}+m \right ) \left (m +2\right ) a \,m^{4} \left (\int _{}^{\frac {9 m \left (\left (m -1\right ) y+3 a \left (\frac {1}{3}+\left (x -\frac {1}{3}\right ) m \right )\right )}{\left (m -1\right ) \left (1+2 m \right ) \left (m +2\right ) \left (-y+a \right )}}\frac {\textit {\_a} {\left (\left (m^{2}+m -2\right ) \textit {\_a} -9 m \right )}^{\frac {1}{m +1}} {\left (\left (2 m^{2}-m -1\right ) \textit {\_a} +9 m \right )}^{\frac {m}{m +1}}}{8 \left (\left (m^{2}-\frac {1}{2} m -\frac {1}{2}\right ) \textit {\_a} +\frac {9 m}{2}\right ) \left (\left (m^{2}+m -2\right ) \textit {\_a} -9 m \right ) {\left (\left (m^{2}+\frac {5}{2} m +1\right ) \textit {\_a} +\frac {9 m}{2}\right )}^{2}}d \textit {\_a} \right )+\left (\frac {1}{3}+\left (x -\frac {1}{3}\right ) m \right ) \left (\frac {m^{2} \left (\left (x -1\right ) a +y\right )}{\left (1+2 m \right ) \left (-y+a \right )}\right )^{\frac {1}{m +1}} \left (-y+a \right ) \left (\frac {\left (a m x +a -y\right ) m}{\left (m +2\right ) \left (-y+a \right )}\right )^{\frac {m}{m +1}}-\frac {2 x \left (\frac {1}{2}+m \right ) \left (m +2\right ) a c_{1} m}{27}\right )}{m \left (2 m^{3}+3 m^{2}-3 m -2\right ) a x} = 0 \]

✗ Solution by Mathematica

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

DSolve[y[x]*D[y[x],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