Internal problem ID [9001]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1422.
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 {2 y}{\sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 57
dsolve(diff(diff(y(x),x),x) = 2/sin(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sin \left (2 x \right )}{-1+\cos \left (2 x \right )}+\frac {c_{2} \left (i \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right ) \sin \left (2 x \right )-2 \cos \left (2 x \right )+2\right )}{-1+\cos \left (2 x \right )} \]
✓ Solution by Mathematica
Time used: 0.07 (sec). Leaf size: 46
DSolve[y''[x] == 2*Csc[x]^2*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\cos (x) \left (c_1-c_2 \log \left (\sqrt {-\sin ^2(x)}-\cos (x)\right )\right )}{\sqrt {-\sin ^2(x)}}-c_2 \\ \end{align*}