55.25.7 problem 7

Internal problem ID [13716]
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 : Sunday, October 12, 2025 at 04:44:30 AM
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*}

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

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\left (\frac {1}{x a -y}\right )^{\frac {1}{n}} \left (\frac {-n y^{2}+\left (2 a n x +b \right ) y-a^{2} n \,x^{2}-a b x +c}{\left (x a -y\right )^{2}}\right )^{-\frac {1}{2 n}} y \,{\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-a b x +b y+2 c}{\sqrt {b^{2}+4 c n}\, \left (-x a +y\right )}\right )}{\sqrt {b^{2}+4 c n}\, n}}-\left (x a -y\right ) x \left (\int _{}^{\frac {1}{x a -y}}\textit {\_a}^{\frac {1}{n}} \left (\textit {\_a}^{2} c -\textit {\_a} b -n \right )^{-\frac {1}{2 n}} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-2 \textit {\_a} c +b}{\sqrt {b^{2}+4 c n}}\right )}{n \sqrt {b^{2}+4 c n}}}d \textit {\_a} a -c_1 \right )}{\left (x a -y\right ) x} = 0 \]

✗ Mathematica

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a**2*n*x**2 + a*b*x - a*x*(2*n + 1)*y(x) - b*y(x) - c + n*y(x)**2 + x*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out