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Solve: y'' + y' cos y = 0

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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#

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