55.24.9 problem 9

Internal problem ID [13638]
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 : 9
Date solved : Sunday, October 12, 2025 at 04:17:48 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 1972

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 0.31 (sec). Leaf size: 200

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

\[ \text {Solve}\left [\frac {a^2 \left (-\frac {2 a \text {arctanh}\left (\frac {a^2-\frac {2 a c (n+1) y(x)}{-b c x^n-b c n x^n+c x^{n+1}}}{a \sqrt {a^2+4 c (n+1)}}\right )}{\sqrt {a^2+4 c (n+1)}}-\log \left (a^2 \left (\frac {a y(x)}{-b c x^n-b c n x^n+c x^{n+1}}+1\right )-\frac {a^2 c (n+1) y(x)^2}{\left (-b c x^n-b c n x^n+c x^{n+1}\right )^2}\right )\right )}{2 c (n+1)}=\frac {a^2 (\log (x-b (n+1))+n \log (x))}{c (n+1)}+c_1,y(x)\right ] \]

✗ 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*x**(n - 1)*(-b*n + x)*y(x) - c*x**(2*n - 1)*(b**2*n*(n + 1) - b*x*(2*n + 1) + x**2) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out