\[
\boxed{
{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left( x \right) =-{\frac { \left( 2\,n+1 \right) \cos \left( x \right) {\frac {\rm d}{{\rm d}x}}y \left( x \right) }{\sin \left( x \right) }}- \left( v+n+1 \right)  \left( v-n \right) y \left( x \right) =0}
\]