Since y' = xz' + z, the equation (H) becomes which is a separable equation. Once solved, go back to the old variable y via the equation y = x z. Let us summarize the steps to follow: (1) Recognize that your equation is an homogeneous equation; that is, you need to check that f(tx,ty)= f(x,y), meaning that f(tx,ty) is independent of the variable t; (2) Write out the substitution z=y/x; (3) Through easy differentiation, find the new equation satisfied by the new function z. You may want to remember the form of the new equation: (4)