Internal problem ID [9759]
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]]
\[ \boxed {y^{\prime \prime }+\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (-2 a +3\right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.703 (sec). Leaf size: 91
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)
\[ y \left (x \right ) = \frac {c_{1} \left (-2+\left (2 a -1\right ) \cos \left (x \right )\right ) \left (2 \cos \left (x \right )+2\right )^{\frac {1}{4}} \sin \left (x \right )^{a -\frac {1}{2}}}{\left (-2 \cos \left (x \right )+2\right )^{\frac {3}{4}}}+\frac {c_{2} \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{-\frac {3}{4}+\frac {a}{2}} \left (\frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}-\frac {a}{2}} \operatorname {hypergeom}\left (\left [a -\frac {1}{2}, -a -\frac {1}{2}\right ], \left [\frac {3}{2}-a \right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (x \right )}} \]
✓ Solution by Mathematica
Time used: 43.509 (sec). Leaf size: 194
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]
\[ y(x)\to \frac {c_1 \sin ^2(x)^{a/2} (-2 a \cos (x)+\cos (x)+2)}{1-\cos (x)}-\frac {c_2 \sin ^2(x)^{-a} (1-\cos (x))^{\frac {a-1}{2}} (\cos (x)+1)^{\frac {a+1}{2}} \left (\frac {(2 a-1) (\cos (x)-1)}{(2 a-1) \cos (x)-2}\right )^{a-\frac {1}{2}} \left (\frac {(2 a-1) (\cos (x)+1)}{(2 a-1) \cos (x)-2}\right )^{a-\frac {1}{2}} \operatorname {AppellF1}\left (2 a,a-\frac {3}{2},a+\frac {1}{2},2 a+1,\frac {3-2 a}{-2 a \cos (x)+\cos (x)+2},\frac {2 a+1}{-2 a \cos (x)+\cos (x)+2}\right )}{4 a^2-2 a} \]