Internal problem ID [9004]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1425.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (-a^{2} \left (\cos ^{2}\relax (x )\right )-\left (3-2 a \right ) \cos \relax (x )-3+3 a \right ) y}{\sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.327 (sec). Leaf size: 99
dsolve(diff(diff(y(x),x),x) = -(-a^2*cos(x)^2-(3-2*a)*cos(x)-3+3*a)/sin(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (-2+\left (2 a -1\right ) \cos \relax (x )\right ) \left (2 \cos \relax (x )+2\right )^{\frac {1}{4}} \left (\sin ^{a -\frac {1}{2}}\relax (x )\right )}{\left (-2 \cos \relax (x )+2\right )^{\frac {3}{4}}}+\frac {c_{2} \hypergeom \left (\left [a -\frac {1}{2}, -\frac {1}{2}-a \right ], \left [\frac {3}{2}-a \right ], \frac {\cos \relax (x )}{2}+\frac {1}{2}\right ) \left (\cos \relax (x )+1\right )^{-\frac {1}{4}-\frac {a}{2}} \left (2 \cos \relax (x )+2\right )^{\frac {3}{4}} \left (\cos \relax (x )-1\right )^{\frac {a}{2}-\frac {1}{4}}}{\left (-2 \cos \relax (x )+2\right )^{\frac {3}{4}}} \]
✓ Solution by Mathematica
Time used: 30.559 (sec). Leaf size: 186
DSolve[y''[x] == (3 - 3*a + (3 - 2*a)*Cos[x] + a^2*Cos[x]^2)*Csc[x]^2*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \sin ^2(x)^{a/2} ((2 a-1) \cos (x)-2)}{\cos (x)-1}-\frac {c_2 \sin ^2(x)^{\frac {1}{2} (-a-1)} (-2 a \cos (x)+\cos (x)+2)^2 \left (\frac {2 a-3}{-2 a \cos (x)+\cos (x)+2}+1\right )^{a+\frac {1}{2}} \left (1-\frac {2 a+1}{-2 a \cos (x)+\cos (x)+2}\right )^{a+\frac {1}{2}} F_1\left (2 a;a-\frac {3}{2},a+\frac {1}{2};2 a+1;\frac {3-2 a}{-2 a \cos (x)+\cos (x)+2},\frac {2 a+1}{-2 a \cos (x)+\cos (x)+2}\right )}{2 a (2 a-1)^3 (\cos (x)-1)} \\ \end{align*}