Internal problem ID [8645]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1065.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 n y^{\prime } \cot \relax (x )+\left (-a^{2}+n^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.066 (sec). Leaf size: 67
dsolve(diff(diff(y(x),x),x)+2*n*diff(y(x),x)*cot(x)+(-a^2+n^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\sin ^{-n +\frac {1}{2}}\relax (x )\right ) \LegendreP \left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \relax (x )\right )+c_{2} \left (\sin ^{-n +\frac {1}{2}}\relax (x )\right ) \LegendreQ \left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.109 (sec). Leaf size: 83
DSolve[(-a^2 + n^2)*y[x] + 2*n*Cot[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (-\sin ^2(x)\right )^{\frac {1}{4}-\frac {n}{2}} \left (c_1 P_{\sqrt {2 n^2-a^2}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))+c_2 Q_{\sqrt {2 n^2-a^2}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))\right ) \\ \end{align*}