I have been researching various methods for solving differential equations. In particular, I want to better understand the factoring approach. For example, if we want to solve a general second order linear ODE of the form $$y''+p(x)y'+q(x)y = 0,$$ We can propse the factorization $$y''+p(x)y'+q(x)y = (D+\phi_1(x))((D+\phi_2(x))y = 0.$$ It turns out that one can find $\phi_1$ and $\phi_2$ by solving an associated Ricatti equation. Then, the original problem may be solved by first solving $$(D+\phi_1(x))Y = 0,$$ and then $$(D+\phi_2(x))y = Y.$$
My question is if this can be extended to higher orders, for example factorizing $$y'''+p(x)y''+q(x)y'+r(x)y = (D+\phi_1)(D+\phi_2)(D+\phi_3)y = 0.$$ If we attempt a similar approach as with the second-order equation, we will arrive at a nonlinear first-order system for $\phi_{1,2,3}$. If curious if there are smart methods for factoring that avoid complicated nonlinear equations.