##### 4.25.20 $$y''(x)=2 y(x) \csc ^2(x)$$

ODE
$y''(x)=2 y(x) \csc ^2(x)$ ODE Classiﬁcation

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.253466 (sec), leaf count = 42

$\left \{\left \{y(x)\to \frac {c_1 \cos (x)}{\sqrt {-\sin ^2(x)}}-\frac {c_2 \cos (x) \sin ^{-1}(\cos (x))}{\sqrt {\sin ^2(x)}}-c_2\right \}\right \}$

Maple
cpu = 0.947 (sec), leaf count = 57

$\left [y \left (x \right ) = \frac {\textit {\_C1} \sin \left (2 x \right )}{-1+\cos \left (2 x \right )}+\frac {\textit {\_C2} \left (-i \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right ) \sin \left (2 x \right )+2 \cos \left (2 x \right )-2\right )}{-1+\cos \left (2 x \right )}\right ]$ Mathematica raw input

DSolve[y''[x] == 2*Csc[x]^2*y[x],y[x],x]

Mathematica raw output

{{y[x] -> -C[2] + (C[1]*Cos[x])/Sqrt[-Sin[x]^2] - (ArcSin[Cos[x]]*C[2]*Cos[x])/S
qrt[Sin[x]^2]}}

Maple raw input

dsolve(diff(diff(y(x),x),x) = 2*y(x)*csc(x)^2, y(x))

Maple raw output

[y(x) = _C1/(-1+cos(2*x))*sin(2*x)+_C2*(-I*ln(cos(2*x)+I*sin(2*x))*sin(2*x)+2*co
s(2*x)-2)/(-1+cos(2*x))]