55.24.65 problem 65

Internal problem ID [13694]
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 : 65
Date solved : Thursday, October 02, 2025 at 02:28:47 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }+\frac {a \left (\frac {\left (3 n +5\right ) x}{2}+\frac {n -1}{n +1}\right ) x^{-\frac {n +4}{n +3}} y}{n +3}&=-\frac {a^{2} \left (\left (n +1\right ) x^{2}-\frac {\left (n^{2}+2 n +5\right ) x}{n +1}+\frac {4}{n +1}\right ) x^{-\frac {n +5}{n +3}}}{2 n +6} \end{align*}

✗ Maple

ode:=y(x)*diff(y(x),x)+a/(3+n)*(1/2*(3*n+5)*x+(n-1)/(n+1))*x^(-(n+4)/(3+n))*y(x) = -a^2/(6+2*n)*((n+1)*x^2-(n^2+2*n+5)/(n+1)*x+4/(n+1))*x^(-(n+5)/(3+n)); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x]*D[y[x],x]+a/(n+3)*((3*n+5)/(2)*x+(n-1)/(n+1))*x^(-(n+4)/(n+3))*y[x]==-a^2/(2*(n+3))*((n+1)*x^2-(n^2+2*n+5)/(n+1)*x+4/(n+1))*x^(-(n+5)/(n+3)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a**2*x**((-n - 5)/(n + 3))*(x**2*(n + 1) - x*(n**2 + 2*n + 5)/(n + 1) + 4/(n + 1))/(2*n + 6) + a*x**((-n - 4)/(n + 3))*(x*(3*n + 5)/2 + (n - 1)/(n + 1))*y(x)/(n + 3) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out