60.3.418 problem 1424

Internal problem ID [11428]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1424
Date solved : Tuesday, January 28, 2025 at 06:05:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.952 (sec). Leaf size: 98

dsolve(sin(x)^2*diff(diff(y(x),x),x)-(a*sin(x)^2+n*(n-1))*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\sqrt {\cos \left (x \right )}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}+\frac {1}{4}} \left (\cos \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{2} +\operatorname {hypergeom}\left (\left [\frac {n}{2}-\frac {i \sqrt {a}}{2}, \frac {n}{2}+\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sin \left (2 x \right )}} \]

✓ Solution by Mathematica

Time used: 0.281 (sec). Leaf size: 65

DSolve[(-((-1 + n)*n) - a*Sin[x]^2)*y[x] + Sin[x]^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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