Internal problem ID [8995]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1416.
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 (2 n +1\right ) \cos \relax (x ) y^{\prime }}{\sin \relax (x )}+\left (v +n +1\right ) \left (v -n \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.075 (sec). Leaf size: 31
dsolve(diff(diff(y(x),x),x) = -(2*n+1)*cos(x)/sin(x)*diff(y(x),x)-(v+n+1)*(v-n)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\sin ^{-n}\relax (x )\right ) \LegendreP \left (v , n , \cos \relax (x )\right )+c_{2} \left (\sin ^{-n}\relax (x )\right ) \LegendreQ \left (v , n , \cos \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.121 (sec). Leaf size: 35
DSolve[y''[x] == (n - v)*(1 + n + v)*y[x] - (1 + 2*n)*Cot[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (-\sin ^2(x)\right )^{-n/2} (c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))) \\ \end{align*}