55.34.10 problem 248

Internal problem ID [14021]
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 : 248
Date solved : Thursday, October 02, 2025 at 09:09:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

Not solved

✗ Sympy

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

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