What do we learn from the Wronskian in the theory of linear ODEs?

For a real interval $$I$$ and a continuous function $$A: I \to \mathbb{R}^{d\times d}$$, let $$(x_1, \dots, x_d)$$ denote a basis of the solution space of the non-autonomous ODE $$\dot x(t) = A(t) x(t) \quad \text{for} \quad t \in I.$$ The mapping $$\varphi: I \ni t \mapsto \det(x_1(t), \dots, x_d(t)) \in \mathbb{R}$$ is usually called the Wronskian of the basis $$(x_1,\dots,x_d)$$, and it seems to be an obligatory topic in every ODE course or book that I've seen.

So in an ODE course that I am currently teaching, I'm facing the following problem:

(1) Despite its prevalence in courses and textbooks, I've rarely (not to say never) encountered any situation where the Wronskian of an ODE is used in a way that sheds non-trivial insight onto the problem at hand - in particular not in any of the books where I've read about it. (Of course, I have also searched on the internet for it, but without any success.)

(2) I feel quite uneasy to teach a concept which I am unable to motivate properly.

(3) I'd feel even more uneasy to just omit it from the course, since chances are that my not knowing of an application of the Wronskian is just due to my ignorance.

Well, what I did is to merely mention the Wronskian in a remark - but of course (and fortunately) I did not get away with it, because quite soon a student asked what the Wronskian is good for.

So this is the

Question: What is the Wronskian (in the context of linear ODEs) good for?

Remarks.

• One can show that $$\varphi$$ satisfies the differential equation $$\dot \varphi(t) = \operatorname{tr}(A(t)) \varphi(t),$$ and since this a one-dimensional equation we have the solution formula $$(*) \qquad \dot \varphi(t) = e^{\int_{t_0}^t \operatorname{tr}(A(s)) \, ds} \varphi(t_0)$$ for it (for any fixed time $$t_0$$ and all $$t \in I$$). This is nice - but still I can't see how to explain to my students that it is useful.

• I've often seen discussions to the end that $$(*)$$ implies that "the Wronskian is non-zero at a time $$t_0$$ if and only if it is non-zero at every time $$t$$" - but I find this somewhat straw man-ish: the fact that $$(x_1(t), \dots, x_d(t))$$ is linearly independent at one time $$t_0$$ if and only if it is linearly independent at every time $$t$$ is an immediate consequence of the uniqueness theorem for ODEs, without any reference to the Wronskian.

• One can give a geometrical interpretation of $$(*)$$: For instance, if all the matrices $$A(t)$$ have trace $$0$$, and it follows that the (non-autonomous) flow associated with our differential equation is volume preserving. However, I'm not convinced that this serves as a sufficient motivation to give the mapping $$t \mapsto \det(x_1(t), \dots, x_d(t))$$ its own name and to discuss it in some detail.

• Maybe a word on the notion "good for" that occurs in the question: I'm pretty comfortable with studying and teaching mathematical objects just in order to better understand them, or for the sake of their intrinsic beauty. However, whenever we do so, this usually happens within a certain theoretical context - i.e., we build a theory, introduce terminology, and this terminology somehow contributes to the development (or to our understanding) of the theory.

Some my question could be rephrased as:

"I'm looking either (i) for applications of the Wronskian of ODEs to concrete problems (within or without mathematics) or (ii) for ways in which the concept 'Wronskian' facilitates our understanding of the theory of ODEs (or of any other theory)."

• The term 'Wronskian' also seems to be used with a more general meaning (see for instance this Wikipedia entry). However, I am specifically interested in the Wronskian for the solutions of a linear ODE.

• The notion of "Evans functions" comes to my mind. Apart from that I have always felt that making this whole Wronskian thing such a big deal in a regular ODE course was mostly due to long habis dying hard, more than real usefulness of the concept. Of course history is important, and perharps there are actually good reasons for studying Wronskians, but I've never really encountered them in any deeply necessary way in my research as applied mathematician/analyst/PDE. So +1 from me, I'm looking forward to enlightning answers! – leo monsaingeon May 27 at 12:46
• The geometric interpretation is quite handy via the liouville's theorem, not just when volumes are preserved (stablilty implies shrinking volumes etc.... ). The geometric picture of how volumes evolve under ODE flow is enlightening, and I believe most ODE students don't really see that picture unless taught using e.g., Arnold's ODE book. Wrronskian is part of that discussion, but not the whole thing ofcourse. – – Piyush Grover May 27 at 13:26
• If $\textrm{tr}\: A=0$, then the Wronskian is a preserved quantity by (*), which is certainly nice to have. In my line of work, it is frequently used in this way (mainly for equations with $d=2$). On the other hand, I don't think there's anything outrageous about not mentioning the Wronskian in an introductory ODE course. – Christian Remling May 27 at 14:46
• In undergraduate texts, Wronskians are usually introduced in the context of second order equations, where all you have to do to prove linear independence of two solutions is show that their ratio is not constant. Unfortunately it seems that this is usually not pointed out to the students. – Michael Renardy May 27 at 21:09
• You might be interested to know that we have a Mathematics Educators Stack Exchange site, although it does not preclude one from asking educational questions on MO or Mathematics Stack Exchange. – J W May 28 at 12:41

Here is a typical use in an undergraduate textbook: to prove that for distinct $$\lambda_j$$ the exponentials $$e^{\lambda_jt}$$ are linearly independent. It has some applications on the more advanced level, but you were asking about undergraduate textbooks. Also notice: uniqueness theorem, even for linear ODE is rarely proved in undergraduate textbooks, at least in the USA. So for linear equations with constant coefficients, the notion of Wronskian permits you to find $$n$$ linearly independent solutions without an appeal to the unproved uniqueness theorem. Same applies to the proof that cosines with distinct frequencies are linearly independent.

Another application. How to write a linear differential equation of order $$n$$ satisfied by $$n$$ given functions $$f_1,\ldots,f_n$$? Here is how: $$\left|\begin{array}{cccc}w&f_1&\ldots&f_n\\ w'&f_1^\prime&\ldots&f_n^\prime\\ \ldots&\ldots&\ldots&\ldots\\ w^{(n)}&f_1^{(n)}&\ldots&f_n^{(n)}\end{array}\right|=0.$$ Expanding with respect to the first column, we obtain that the Wronskian $$W=W(f_1,\ldots,f_n)$$ is the coefficient at $$w^{(n)}$$, in particular, if all $$f_j$$ are analytic then the singular points of the equation are the zeros of $$W$$.

The significance of the Wronskian is not limited to differential equations. Consider a finite-dimensional vector space $$V$$ consisting of functions. (For example, polynomials of degree at most $$n$$). Suppose we have a basis $$f_1,\ldots,f_n$$. How to expand a function $$f\in V$$ in this basis? Write $$f=c_1f_1+\ldots+c_nf_n,$$ differentiate $$n-1$$ times and solve the linear system with respect to $$c_j$$. The determinant of this system is the Wronskian. This was the original goal of Heine-Wronski when he invented it.

For less elementary applications, type "Wronski map" in the cell "Anywhere" or in the cell "Title" in Mathscinet search.

• See my answer here for a simple inductive proof that the exponentials are linearly independent, without any appeal to Wronskians. – Robert Israel May 27 at 20:22
• +1 Thank you very much your answer! Although I tend to agree with @RobertIsrael that his "direct" proof is a bit more natural than a proof that uses the Wronskian, I find this answer very useful anyway: a few such facts which can also be proved with appeal to the Wronskian provide at least some context for the Wronskian. (And context is probably one of the most important things in order to motivate a mathematical concept.) – Jochen Glueck May 27 at 21:23
• Concerning the uniqueness theorem: Indeed, the point that the Wronskian allows us to deal with linear ODEs without appeal to the uniqueness theorem also briefly occurred to me. But I think this is mostly a point about the academic system: in universities in Germany (where I teach) the existence and uniqueness theorem is typically taught (and proved) in most introductory ODE courses. (I'd guess it's similar in many other European countries, but I should probably be careful not to extrapolate too much.) – Jochen Glueck May 27 at 21:31
• @Jochen Glueck: sorry I read another proof in the link given by Robert Israel, the first one. – Alexandre Eremenko May 28 at 12:34
• I really like all the answers, so I couldn't make up my mind which one to accept - hence, I finally accepted this answer since it was the first one that was posted. – Jochen Glueck Jun 3 at 16:32

Quite an important use of the Wronskian arises in the spectral analysis of the Hill operator $$\frac{d^2}{dx^2}+q(x)$$ when $$q$$ is periodic. This is the search of Floquet exponents.

• +1 Very interesting answer, thanks a lot! (Embarrasingly, Floquet theory has not been on my radar at all.) – Jochen Glueck May 27 at 21:41
• Curiously, I don't recall Arscott ever discussing the Wronskian in his treatise. (I have still upvoted this answer.) – J. M. isn't a mathematician May 28 at 1:19

This is in the same spirit as Piyush Grover's comment. The determinant $$\det(x_1(t),\ldots,x_n(t))$$ definitely deserves a name (not only in the context of linear ODE's). In such a lecture students could (and ,in my opinion, should) learn the meaning of the divergence of a vector field $$F$$. Having learned Picard-Lindelöf they are ready to understand the flow $$\phi(t,x)$$ as he solution of the initial value Problem $$\phi'(t,x)=F(\phi(t,x))$$, $$\phi'(0,x)=x$$, and for a small cube $$x+[0,r]^n$$ you can throw the edges into the flow to get after a short time almost the parallelepiped with edges $$\phi(t,x+re_j)-\phi(t,x)$$ whose (oriented) volume compared to the volume of the cube is $$v(t,r)=\det[\phi(t,x+re_1)-\phi(t,x),\ldots,\phi(t,r+e_n)-\phi(t,x)]/r^n$$ If you take the derivative $$\partial_t$$ at $$0$$ and the limit $$r\to 0$$ you get the divergence of the vector field (no problem to take time dependent vector fields).

The desire to make this more precise also motivates the theorems about differentiability of the solutions of initial value problems with respect to the initial values. Then you can throw quite arbitrary small sets into the flow and compare the evolved (oriented) volume with the original one by calculating them with the $$n$$-dimensional substitution rule.

• Thanks a lot for your answer! (+1) I like this geometrical point of view very much - so you nearly convinced me to include this kind of reasoning the next time when I teach such a course (for the current semester, the opportunity is most likely gone since it would have required me to organise the contents of the course in a somewhat different way than how I have actually done it). – Jochen Glueck May 30 at 11:43