Internal problem ID [9009]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1430.
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 {\cos \relax (x ) y^{\prime }}{\sin \relax (x )}+\frac {\left (v \left (v +1\right ) \left (\sin ^{2}\relax (x )\right )-n^{2}\right ) y}{\sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.107 (sec). Leaf size: 96
dsolve(diff(diff(y(x),x),x) = -1/sin(x)*cos(x)*diff(y(x),x)-(v*(v+1)*sin(x)^2-n^2)/sin(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {v}{2}+\frac {n}{2}, \frac {1}{2}+\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}}+c_{2} \sqrt {\cos \left (2 x \right )+1}\, \hypergeom \left (\left [1+\frac {v}{2}+\frac {n}{2}, \frac {1}{2}-\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}} \]
✓ Solution by Mathematica
Time used: 0.271 (sec). Leaf size: 22
DSolve[y''[x] == -(Csc[x]^2*(-n^2 + v*(1 + v)*Sin[x]^2)*y[x]) - Cot[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x)) \\ \end{align*}