2.34.14 Problem 252

Internal problem ID [13912]
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 : 252
Date solved : Friday, December 19, 2025 at 08:28:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

2.34.14.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 61

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

\[ y = c_1 \,{\mathrm e}^{i b \int x^{n -1} \sqrt {-\frac {1}{x^{2 n}+a}}d x}+c_2 \,{\mathrm e}^{-i b \int x^{n -1} \sqrt {-\frac {1}{x^{2 n}+a}}d x} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
   Solution is available but has integrals. Trying a simpler solution using Kov\ 
acics algorithm... 
   Solution via Kovacic is not simpler. Returning default solution 
   <- linear_1 successful
 

2.34.14.2 ✓ Mathematica. Time used: 1.165 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to c_1 \cosh \left (\frac {b \text {arcsinh}\left (\frac {x^n}{\sqrt {a}}\right )}{n}\right )+i c_2 \sinh \left (\frac {b \text {arcsinh}\left (\frac {x^n}{\sqrt {a}}\right )}{n}\right ) \end{align*}

2.34.14.3 ✗ Sympy

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

TypeError : Add 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')