61.25.7 problem 7

Internal problem ID [12500]
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 1.3.4-2.
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 08:01:53 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x y y^{\prime }&=-n y^{2}+a \left (2 n +1\right ) x y+b y-a^{2} n \,x^{2}-a b x +c \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 224

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

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

✗ Solution by Mathematica

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

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

Not solved