60.3.410 problem 1416

Internal problem ID [11420]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1416
Date solved : Tuesday, January 28, 2025 at 06:05:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y \end{align*}

✓ Solution by Maple

Time used: 0.391 (sec). Leaf size: 26

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)
 

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

✓ Solution by Mathematica

Time used: 0.191 (sec). Leaf size: 35

DSolve[D[y[x],{x,2}] == (n - v)*(1 + n + v)*y[x] - (1 + 2*n)*Cot[x]*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (-\sin ^2(x)\right )^{-n/2} (c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))) \]