55.24.13 problem 13

Internal problem ID [13642]
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 : 13
Date solved : Wednesday, October 01, 2025 at 10:05:35 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=\left (a \left (1+2 n \right ) x^{2}+c x +b \left (2 n -1\right )\right ) x^{-2+n} y-\left (n \,a^{2} x^{4}+a c \,x^{3}+n \,b^{2}+b c x +d \,x^{2}\right ) x^{2 n -3} \end{align*}

✗ Maple

ode:=y(x)*diff(y(x),x) = (a*(2*n+1)*x^2+c*x+b*(-1+2*n))*x^(n-2)*y(x)-(a^2*n*x^4+a*c*x^3+b^2*n+b*c*x+d*x^2)*x^(2*n-3); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x]*D[y[x],x]==(a*(2*n+1)*x^2+c*x+b*(2*n-1))*x^(n-2)*y[x]-(n*a^2*x^4+a*c*x^3+d*x^2+b*c*x+n*b^2)*x^(2*n-3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-x**(n - 2)*(a*x**2*(2*n + 1) + b*(2*n - 1) + c*x)*y(x) + x**(2*n - 3)*(a**2*n*x**4 + a*c*x**3 + b**2*n + b*c*x + d*x**2) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out