55.34.9 problem 247

Internal problem ID [14020]
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 : 247
Date solved : Thursday, October 02, 2025 at 09:09:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -(-a**2*b*x*y(x) - x**n*Derivative(y(x), (x, 2)))/(a*b*x + a*x**n + b - x**(n - 1)) + Derivative(y(x), x) cannot be solved by the factorable group method