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.


1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$
    – Doug
    Dec 24, 2021 at 16:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.