55.34.8 problem 246

Internal problem ID [14019]
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 : 246
Date solved : Thursday, October 02, 2025 at 09:09:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

ode:=x^n*diff(diff(y(x),x),x)+(a*x^n+b)*diff(y(x),x)+c*((a-c)*x^n+b)*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+b)*D[y[x],x]+c*((a-c)*x^n+b)*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") 
c = symbols("c") 
n = symbols("n") 
y = Function("y") 
ode = Eq(c*(b + x**n*(a - c))*y(x) + x**n*Derivative(y(x), (x, 2)) + (a*x**n + b)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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