60.3.64 problem 1065

Internal problem ID [11074]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1065
Date solved : Tuesday, January 28, 2025 at 05:40:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.618 (sec). Leaf size: 60

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)
 

\[ y = \sin \left (x \right )^{-n +\frac {1}{2}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \left (x \right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.188 (sec). Leaf size: 83

DSolve[(-a^2 + n^2)*y[x] + 2*n*Cot[x]*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (-\sin ^2(x)\right )^{\frac {1}{4}-\frac {n}{2}} \left (c_1 P_{\sqrt {2 n^2-a^2}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))+c_2 Q_{\sqrt {2 n^2-a^2}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))\right ) \]