ODE
\[ y''(x)=2 y(x) \csc ^2(x) \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0846629 (sec), leaf count = 54
\[\left \{\left \{y(x)\to \frac {\cos (x) \left (c_2 \log \left (\sqrt {-\sin ^2(x)}+\cos (x)\right )+c_1\right )-c_2 \sqrt {-\sin ^2(x)}}{\sqrt {-\sin ^2(x)}}\right \}\right \}\]
Maple ✓
cpu = 0.248 (sec), leaf count = 50
\[ \left \{ y \left ( x \right ) ={\frac {i\ln \left ( \cos \left ( 2\,x \right ) +i\sin \left ( 2\,x \right ) \right ) \sin \left ( 2\,x \right ) {\it \_C2}+{\it \_C1}\,\sin \left ( 2\,x \right ) -2\,{\it \_C2}\, \left ( -1+\cos \left ( 2\,x \right ) \right ) }{-1+\cos \left ( 2\,x \right ) }} \right \} \] Mathematica raw input
DSolve[y''[x] == 2*Csc[x]^2*y[x],y[x],x]
Mathematica raw output
{{y[x] -> (Cos[x]*(C[1] + C[2]*Log[Cos[x] + Sqrt[-Sin[x]^2]]) - C[2]*Sqrt[-Sin[x]^2])/Sqrt[-Sin[x]^2]}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = 2*y(x)*csc(x)^2, y(x),'implicit')
Maple raw output
y(x) = (I*ln(cos(2*x)+I*sin(2*x))*sin(2*x)*_C2+_C1*sin(2*x)-2*_C2*(-1+cos(2*x)))/(-1+cos(2*x))