60.3.67 problem 1068

Internal problem ID [11077]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1068
Date solved : Tuesday, January 28, 2025 at 05:40:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime } \cot \left (x \right )+v \left (v +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.744 (sec). Leaf size: 45

dsolve(diff(diff(y(x),x),x)+diff(y(x),x)*cot(x)+v*(v+1)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [-\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+c_{2} \cos \left (x \right ) \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.148 (sec). Leaf size: 20

DSolve[v*(1 + v)*y[x] + Cot[x]*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {LegendreP}(v,\cos (x))+c_2 \operatorname {LegendreQ}(v,\cos (x)) \]