2.34.15 Problem 253

Internal problem ID [13913]
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 : 253
Date solved : Friday, December 19, 2025 at 08:29:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.34.15.1 ✓ Maple. Time used: 0.020 (sec). Leaf size: 23

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

\[ y = c_1 \operatorname {LegendreP}\left (\nu , a \,x^{n}\right )+c_2 \operatorname {LegendreQ}\left (\nu , a \,x^{n}\right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
   -> Trying a Liouvillian solution using Kovacics algorithm 
   <- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   <- Legendre successful 
<- special function solution successful
 

2.34.15.2 ✓ Mathematica. Time used: 0.074 (sec). Leaf size: 79

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

\begin{align*} y(x)&\to i a c_2 \sqrt {x^{2 n}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {\nu }{2},\frac {\nu }{2}+1,\frac {3}{2},a^2 x^{2 n}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {\nu }{2},\frac {\nu +1}{2},\frac {1}{2},a^2 x^{2 n}\right ) \end{align*}

2.34.15.3 ✗ Sympy

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

TypeError : Mul object cannot be interpreted as an integer
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_hypergeometric', '2nd_hypergeometric_Integral', '2nd_power_series_regular')