set z = y' = dy/dx
thus, y'' = dz/dx = dz/dy dy/dx = (dz/dy) y' = (dz/dy)z
Thus, the above equation becomes a first-order differential equation of z (dependent variable) with respect to y (independent variable):
dz/dy = z + z3 cos y = 0
which can be solved by separation of variables:
- dz/z2 = cos y dy or 1/z = sin y + c1
or z = y' = dy/dx = 1/(sin y + c1)
which can be solved by separation of variables again
(sin y + c1) dy = dx ⇒ - cos y + c1 y + c2 = x#