Internal problem ID [9007]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1428.
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 \left (\cos ^{2}\relax (x )\right )+\left (\sin ^{2}\relax (x )\right ) b +c \right ) y}{\sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 219
dsolve(diff(diff(y(x),x),x) = -(a*cos(x)^2+b*sin(x)^2+c)/sin(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {1}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {-2 \cos \left (2 x \right )+2}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {-4 a +1-4 c}}{4}}}{\sqrt {\sin \left (2 x \right )}}+\frac {c_{2} \left (2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {3}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {-2 \cos \left (2 x \right )+2}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {-4 a +1-4 c}}{4}}}{\sqrt {\sin \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.25 (sec). Leaf size: 87
DSolve[y''[x] == -(Csc[x]^2*(c + a*Cos[x]^2 + b*Sin[x]^2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{\sqrt {b-a}-\frac {1}{2}}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))+c_2 Q_{\sqrt {b-a}-\frac {1}{2}}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))\right ) \\ \end{align*}