Internal problem ID [9003]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1424.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } \left (\sin ^{2}\relax (x )\right )-\left (a \left (\sin ^{2}\relax (x )\right )+n \left (n -1\right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 125
dsolve(sin(x)^2*diff(diff(y(x),x),x)-(a*sin(x)^2+n*(n-1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}} \hypergeom \left (\left [\frac {n}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\frac {c_{2} \left (2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (-2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}}}{\sqrt {\sin \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 65
DSolve[(-((-1 + n)*n) - a*Sin[x]^2)*y[x] + Sin[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{i \sqrt {a}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))+c_2 Q_{i \sqrt {a}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))\right ) \\ \end{align*}