# Factoring higher-order differential operators

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 $$y′′+p(x)y′+q(x)y=\big(D+\phi_1(x)\big)\big((D+\phi_2(x)\big)y=0.$$ It turns out that one can find $$\phi_1$$ and $$\phi_2$$ by solving an associated Riccati equation. Then, the original problem may be solved by first solving $$\big(D+\phi_1(x)\big)Y=0,$$ and then $$\big(D+\phi_2(x)\big)y=Y.$$ 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 $$\phi_1,\phi_2, \phi_3$$. If curious if there are smart methods for factoring that avoid complicated nonlinear equations.

I'm not sure why you refer to a first order non-linear equation in relation to the factorization of a third order operator. Expanding both sides of the desired factorization identity, you get in the second order case \begin{aligned} \phi_1 + \phi_2 &= p , \\ \phi_2' + \phi_1 \phi_2 &= q . \end{aligned} Indeed, the first equation for $$\phi_1$$ and substituting it into the second one you get a first order Ricatti equation for $$\phi_2$$. However, in the third order case, the analogous system is \begin{aligned} \phi_1 + \phi_2 + \phi_3 &= p , \\ \phi_2' + \phi_1 \phi_2 + (\phi_1+\phi_2) \phi_3 + 2\phi_3' &= q , \\ \phi_3'' + (\phi_1+\phi_2) \phi_3' + \phi_3 \phi_2' + \phi_1 \phi_2 \phi_3 &= r . \end{aligned} Here, you can still eliminate $$\phi_1$$ using the first equation, as well as eliminate $$\phi_2'$$ from the third equation using the second equation. The result is a mixed order (first order in $$\phi_2$$ and second order in $$\phi_3$$) non-linear (with quadratic and cubic nonlinearities) for $$\phi_2$$ and $$\phi_3$$.

If you impose no restrictions on the coefficients $$\phi_{1,2,3}$$ or $$p,q,r$$, beyond smoothness say, then such a factorization is always possible and the possibilities are fully described by the existence and uniqueness theory for the nonlinear ODE system on the $$\phi_i$$. Since factorization is an inherently nonlinear process, I doubt that you can get better statement than that.

If, on the other hand, you impose some algebraic restriction on the coefficients, say requiring that the $$\phi_i$$ and $$p,q,r$$ functions must all be polynomial, or rational, or analytic, then you can do better. In fact, then there are some algorithms that can decide whether a factorization exists or not. For instance, the following reference deals with the case of rational coefficients:

van Hoeij, Mark, Factorization of differential operators with rational functions coefficients, J. Symb. Comput. 24, No. 5, 537-561 (1997). ZBL0886.68082.

A slightly more up to date discussion, with more references is also in §4.2 of

van der Put, Marius; Singer, Michael F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften 328. Berlin: Springer (ISBN 3-540-44228-6/hbk). xvii, 438 p. (2003). ZBL1036.12008.

• Thank you! I was just guessing when I said nonlinear first order system. I hadn't worked it out completely when I posted this question.
– Doug
Dec 24, 2021 at 16:51