Internal problem ID [9015]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1436.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (4 v \left (v +1\right ) \left (\sin ^{2}\relax (x )\right )-\left (\cos ^{2}\relax (x )\right )+2-4 n^{2}\right ) y}{4 \sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.146 (sec). Leaf size: 140
dsolve(diff(diff(y(x),x),x) = -1/4*(4*v*(v+1)*sin(x)^2-cos(x)^2+2-4*n^2)/sin(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {v}{2}+\frac {n}{2}, \frac {1}{2}+\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}} \sqrt {-2 \cos \left (2 x \right )+2}}{\sqrt {\sin \left (2 x \right )}}+\frac {c_{2} \left (2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} \hypergeom \left (\left [1+\frac {v}{2}+\frac {n}{2}, \frac {1}{2}-\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}} \sqrt {-2 \cos \left (2 x \right )+2}}{\sqrt {\sin \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.335 (sec). Leaf size: 33
DSolve[y''[x] == -1/4*(Csc[x]^2*(2 - 4*n^2 - Cos[x]^2 + 4*v*(1 + v)*Sin[x]^2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [4]{-\sin ^2(x)} (c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))) \\ \end{align*}