61.29.13 problem 122

Internal problem ID [12543]
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 : 122
Date solved : Monday, March 31, 2025 at 05:38:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{2 n} \left (b \,x^{n}+c \right )^{m}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y&=0 \end{align*}

✗ Maple

ode:=x^2*diff(diff(y(x),x),x)+(a*x^(2*n)*(b*x^n+c)^m+1/4-1/4*n^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=x^2*D[y[x],{x,2}]+(a*x^(2*n)*(b*x^n+c)^m+1/4-1/4*n^2)*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") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (a*x**(2*n)*(b*x**n + c)**m - n**2/4 + 1/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Not sure of sign of -2*n