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 secondorder equation, we will arrive at a nonlinear firstorder system for $\phi_1,\phi_2, \phi_3$. If curious if there are smart methods for factoring that avoid complicated nonlinear equations.
1 Answer
I'm not sure why you refer to a first order nonlinear 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 $\phi_3$) nonlinear (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, 537561 (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 3540442286/hbk). xvii, 438 p. (2003). ZBL1036.12008.

$\begingroup$ 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. $\endgroup$– DougDec 24, 2021 at 16:51