Question

solution for dy/dx+xsin 2y=x^3 cos^2y

Answer 1

start by putting the dy/dx by itself
$\frac{dy}{dx}+xsin2y=x^3 cos^2y \\ \frac{dy}{dx}=x^3 cos^2y-xsin2y$
then multiply both sides by dx
$dy=(x^3cos^2y-xsin2y)dx$
which you can now integrate both sides with respect to their variable, so the left by y and the right by x
$y+ c_{1}= \frac{x^4}{4}cos^2y-\frac{x^2}{2}sin2y+c_{2}$
let c = c2-c1 and you find
$y= \frac{x^4}{4}cos^2y-\frac{x^2}{2}sin2y+c$