Question

solve it with sepration. 8cos^2ydx+cosec^2xdy=0

Answer 1

\[\begin{align*}
8\cos^2y\,dx+\csc^2x\,dy&=0\\\\
\sec^2y\,dy&=-8\sin^2x\,dx
\end{align*}\]

Answer 2

When integrating, you might find this identity useful:
\[\sin^2x=\frac{1}{2}-\frac{1}{2}\cos2x\]

Answer 3

Another way of solving this is by finding an appropriate integrating factor. Notice that multiplying both sides of the ODE by \(\dfrac{\sin^2x}{\cos^2y}\) yields an exact equation,
\[8\sin^2x\,dx+\sec^2y\,dy=0\]
which gives the same result.