61.29.39 problem 148

Internal problem ID [12569]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 148
Date solved : Monday, March 31, 2025 at 05:39:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=x^2*D[y[x],{x,2}]+(a*x^(n+2)+b*x^2+c)*D[y[x],x]+(a*n*x^(n+1)+a*c*x^n+b*c)*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(x**2*Derivative(y(x), (x, 2)) + (a*x**(n + 2) + b*x**2 + c)*Derivative(y(x), x) + (a*c*x**n + a*n*x**(n + 1) + b*c)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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