2.33.27 Problem 236

Internal problem ID [13897]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 236
Date solved : Friday, December 19, 2025 at 08:20:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.33.27.1 ✓ Maple. Time used: 0.048 (sec). Leaf size: 68

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

\[ y = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {m^{2}}{4}-\frac {\lambda }{4}, -x^{2}\right ) c_2 x +\operatorname {HeunC}\left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {m^{2}}{4}-\frac {\lambda }{4}, -x^{2}\right ) c_1 \right ) \left (x^{2}+1\right )^{\frac {m}{2}} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
<- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Whittaker 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
   -> hypergeometric 
      -> heuristic approach 
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
   -> Mathieu 
      -> Equivalence to the rational form of Mathieu ODE under a power @ Moebi\ 
us 
trying a solution in terms of MeijerG functions 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
<- Heun successful: received ODE is equivalent to the  HeunC  ODE, case  a <> 0\ 
, e <> 0, c = 0
 

2.33.27.2 ✓ Mathematica. Time used: 0.207 (sec). Leaf size: 124

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

\begin{align*} y(x)&\to \left (x^2+1\right )^{\frac {\sqrt {m^2}}{2}} \left (c_2 x \text {HeunC}\left [\frac {1}{4} \left (\lambda -m^2-3 \sqrt {m^2}-2\right ),-\frac {a^2}{4},\frac {3}{2},\sqrt {m^2}+1,0,-x^2\right ]+c_1 \text {HeunC}\left [\frac {1}{4} \left (\lambda -m^2-\sqrt {m^2}\right ),-\frac {a^2}{4},\frac {1}{2},\sqrt {m^2}+1,0,-x^2\right ]\right ) \end{align*}

2.33.27.3 ✗ Sympy

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

False
 

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_power_series_ordinary')