61.24.56 problem 56

Internal problem ID [12469]
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 : 56
Date solved : Tuesday, January 28, 2025 at 07:59:51 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y-\frac {a \left (\left (k +1\right ) x -1\right ) y}{x^{2}}&=\frac {a^{2} \left (k +1\right ) \left (x -1\right )}{x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 143

dsolve(y(x)*diff(y(x),x)-a*((k+1)*x-1)*x^(-2)*y(x)=a^2*(k+1)*(x-1)*x^(-2),y(x), singsol=all)
 

\[ \frac {\left (\frac {a x}{-x y+a}\right )^{-\frac {1}{k +1}} x^{2} \left (\frac {\left (x -1\right ) a +x y}{-x y+a}\right )^{\frac {1}{k +1}} {\mathrm e}^{\frac {-x y+a}{a \left (k +1\right ) x}} y-\left (\int _{}^{\frac {a x}{-x y+a}}\left (\textit {\_a} -1\right )^{\frac {1}{k +1}} {\mathrm e}^{\frac {1}{\left (k +1\right ) \textit {\_a}}} \textit {\_a}^{-\frac {1}{k +1}}d \textit {\_a} -c_{1} \right ) \left (-x y+a \right )}{-x y+a} = 0 \]

✗ Solution by Mathematica

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

DSolve[y[x]*D[y[x],x]-a*((k+1)*x-1)*x^(-2)*y[x]==a^2*(k+1)*(x-1)*x^(-2),y[x],x,IncludeSingularSolutions -> True]
 

Not solved