60.3.361 problem 1367

Internal problem ID [11371]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1367
Date solved : Tuesday, January 28, 2025 at 06:04:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.946 (sec). Leaf size: 88

dsolve(diff(diff(y(x),x),x) = -2/(x^2+1)*x*diff(y(x),x)-(a^2*(x^2+1)^2-n*(n+1)*(x^2+1)+m^2)/(x^2+1)^2*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.355 (sec). Leaf size: 140

DSolve[D[y[x],{x,2}] == -(((m^2 - n*(1 + n)*(1 + x^2) + a^2*(1 + x^2)^2)*y[x])/(1 + x^2)^2) - (2*x*D[y[x],x])/(1 + x^2),y[x],x,IncludeSingularSolutions -> True]
 

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