55.34.22 problem 260

Internal problem ID [14033]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 260
Date solved : Thursday, October 02, 2025 at 09:09:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{n +1}+b \,x^{n}+c \right )^{2} y^{\prime \prime }+\left (\alpha \,x^{n}+\beta \,x^{n -1}+\gamma \right ) y^{\prime }+\left (n \left (-a n -a +\alpha \right ) x^{n -1}+\left (n -1\right ) \left (-b n +\beta \right ) x^{-2+n}\right ) y&=0 \end{align*}

✗ Maple

ode:=(x^(n+1)*a+b*x^n+c)^2*diff(diff(y(x),x),x)+(alpha*x^n+beta*x^(n-1)+gamma)*diff(y(x),x)+(n*(-a*n-a+alpha)*x^(n-1)+(n-1)*(-b*n+beta)*x^(n-2))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=(a*x^(n+1)+b*x^n+c)^2*D[y[x],{x,2}]+(\[Alpha]*x^n+\[Beta]*x^(n-1)+\[Gamma])*D[y[x],x]+(n*(\[Alpha]-a-a*n)*x^(n-1)+(n-1)*(\[Beta]-b*n)*x^(n-2))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer