Internal problem ID [8999]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1420.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (\cos ^{2}\relax (x )\right ) y^{\prime \prime }-\left (a \left (\cos ^{2}\relax (x )\right )+n \left (-1+n \right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.151 (sec). Leaf size: 123
dsolve(cos(x)^2*diff(diff(y(x),x),x)-(a*cos(x)^2+n*(n-1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \left (2 x \right ) \left (\cos ^{-n}\relax (x )\right ) \hypergeom \left (\left [1+\frac {i \sqrt {a}}{2}-\frac {n}{2}, 1-\frac {i \sqrt {a}}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}-n \right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\frac {c_{2} \left (-2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [n +\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \left (\cos ^{n}\relax (x )\right )}{\sqrt {\sin \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.271 (sec). Leaf size: 126
DSolve[(-((-1 + n)*n) - a*Cos[x]^2)*y[x] + Cos[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 i^{1-n} \cos ^{1-n}(x) \, _2F_1\left (\frac {1}{2} \left (-n-i \sqrt {a}+1\right ),\frac {1}{2} \left (-n+i \sqrt {a}+1\right );\frac {3}{2}-n;\cos ^2(x)\right )+c_2 i^n \cos ^n(x) \, _2F_1\left (\frac {1}{2} \left (n-i \sqrt {a}\right ),\frac {1}{2} \left (n+i \sqrt {a}\right );n+\frac {1}{2};\cos ^2(x)\right ) \\ \end{align*}