61.32.26 problem 235

Internal problem ID [12736]
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 : Tuesday, January 28, 2025 at 08:23:55 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*}

✓ Solution by Maple

Time used: 1.542 (sec). Leaf size: 64

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

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

✓ Solution by Mathematica

Time used: 0.358 (sec). Leaf size: 234

DSolve[(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,y[x],x,IncludeSingularSolutions -> True]
 

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