55.33.26 problem 235

Internal problem ID [14009]
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 : 235
Date solved : Friday, October 03, 2025 at 07:23:25 AM
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*}

✓ Maple. Time used: 0.094 (sec). Leaf size: 64

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);
 

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

✓ Mathematica. Time used: 0.218 (sec). Leaf size: 234

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 e^{i \sqrt {a^2} x} \left (\frac {x+1}{x-1}\right )^{\frac {\sqrt {m^2}}{2}} \left (c_2 (x-1)^{\sqrt {m^2}} \text {HeunC}\left [-\left (\sqrt {m^2}+1\right ) \left (\sqrt {m^2}+2 i \sqrt {a^2}\right )-a^2+\lambda ,-4 i \sqrt {a^2} \left (\sqrt {m^2}+1\right ),\sqrt {m^2}+1,\sqrt {m^2}+1,-4 i \sqrt {a^2},\frac {1-x}{2}\right ]+c_1 \text {HeunC}\left [2 i \sqrt {a^2} \left (\sqrt {m^2}-1\right )-a^2+\lambda ,-4 i \sqrt {a^2},1-\sqrt {m^2},\sqrt {m^2}+1,-4 i \sqrt {a^2},\frac {1-x}{2}\right ]\right ) \end{align*}

✗ 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