61.32.27 problem 236

Internal problem ID [12737]
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 : Tuesday, January 28, 2025 at 08:23:56 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: 1.554 (sec). Leaf size: 68

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: 0.342 (sec). Leaf size: 124

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]
 

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