Note 1:

sometimes a second order D.E. can be reduced into a first order by substituting $v=q(D_x y) \Rightarrow D_x v=w(D_x^2 y)$. (e.g. $x D_x^2 y + D_xy = Q(x)$ sub $v= D_x y \Rightarrow D_x v = D_x^2 y \Rightarrow x D_x v + v = Q(x)$ is 1st order)