60.3.424 problem 1430

Internal problem ID [11434]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1430
Date solved : Tuesday, January 28, 2025 at 06:05:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.678 (sec). Leaf size: 79

dsolve(diff(diff(y(x),x),x) = -1/sin(x)*cos(x)*diff(y(x),x)-(v*(v+1)*sin(x)^2-n^2)/sin(x)^2*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.488 (sec). Leaf size: 22

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

\[ y(x)\to c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x)) \]