Today’s is another topic suggested by Mr Wu, author of the Singapore Maths Tuition blog. The Wronskian is named for Józef Maria Hoëne-Wroński, a Polish mathematician, born in 1778. He served in General Tadeusz Kosciuszko’s army in the 1794 Kosciuszko Uprising. After being captured and forced to serve in the Russian army, he moved to France. He kicked around Western Europe and its mathematical and scientific circles. I’d like to say this was all creative and insightful, but, well. Wikipedia describes him trying to build a perpetual motion machine. Trying to square the circle (also impossible). Building a machine to predict the future. The St Andrews mathematical biography notes his writing a summary of “the general solution of the fifth degree [polynomial] equation”. This doesn’t exist.
Both sources, though, admit that for all that he got wrong, there were flashes of insight and brilliance in his work. The St Andrews biography particularly notes that Wronski’s tables of logarithms were well-designed. This is a hard thing to feel impressed by. But it’s hard to balance information so that it’s compact yet useful. He wrote about the Wronskian in 1812; it wouldn’t be named for him until 1882. This was 29 years after his death, but it does seem likely he’d have enjoyed having a familiar thing named for him. I suspect he wouldn’t enjoy my next paragraph, but would enjoy the fight with me about it.
The Wronskian is a thing put into Introduction to Ordinary Differential Equations courses because students must suffer in atonement for their sins. Those who fail to reform enough must go on to the Hessian, in Partial Differential Equations.
To be more precise, the Wronskian is the determinant of a matrix. The determinant you find by adding and subtracting products of the elements in a matrix together. It’s not hard, but it is tedious, and gets more tedious pretty fast as the matrix gets bigger. (In Big-O notation, it’s the order of the cube of the matrix size. This is rough, for things humans do, although not bad as algorithms go.) The matrix here is made up of a bunch of functions and their derivatives. The functions need to be ones of a single variable. The derivatives, you need first, second, third, and so on, up to one less than the number of functions you have.
If you have two functions, and , you need their first derivatives, and . If you have three functions, , , and , you need first derivatives, , , and , as well as second derivatives, , , and . If you have functions and here I’ll call them , you need derivatives, and so on through . You see right away this is a fun and exciting thing to calculate. Also why in intro to differential equations you only work this out with two or three functions. Maybe four functions if the class has been really naughty.
Go through your functions and your derivatives and make a big square matrix. And then you go through calculating the derivative. This involves a lot of multiplying strings of these derivatives together. It’s a lot of work. But at least doing all this work gets you older.
So one will ask why do all this? Why fit it into every Intro to Ordinary Differential Equations textbook and why slip it in to classes that have enough stuff going on?
One answer is that if the Wronskian is not zero for some values of the independent variable, then the functions that went into it are linearly independent. Mathematicians learn to like sets of linearly independent functions. We can treat functions like directions in space. Linear independence assures us none of these functions are redundant, pointing a way we already can describe. (Real people see nothing wrong in having north, east, and northeast as directions. But mathematicians would like as few directions in our set as possible.) The Wronskian being zero for every value of the independent variable seems like it should tell us the functions are linearly dependent. It doesn’t, not without some more constraints on the functions.
This is fine, but who cares? And, unfortunately, in Intro it’s hard to reach a strong reason to care. To this major, the emphasis on linearly independent functions felt misplaced. It’s the sort of thing we care about in linear algebra. Or some course where we talk about vector spaces. Differential equations do lead us into vector spaces. It’s hard to find a corner of analysis that doesn’t.
Every ordinary differential equation has a secret picture. This is a vector field. One axis in the field is the independent variable of the function. The other axes are the value of the function. And maybe its derivatives, depending on how many derivatives are used in the ordinary differential equation. To solve one particular differential equation is to find one path in this field. People who just use differential equations will want to find one path.
Mathematicians tend to be fine with finding one path. But they want to find what kinds of paths there can be. Are there paths which the differential equation picks out, by making paths near it stay near? Or by making paths that run away from it? And here is the value of the Wronskian. The Wronskian tells us about the divergence of this vector field. This gives us insight to how these paths behave. It’s in the same way that knowing where high- and low-pressure systems are describes how the weather will change. The Wronskian, by way of a thing called Liouville’s Theorem that I haven’t the strength to describe today, ties in to the Hamiltonian. And the Hamiltonian we see in almost every mechanics problem of note.
You can see where the mathematics PhD, or the physicist, would find this interesting. But what about the student, who would look at the symbols evoked by those paragraphs above with reasonable horror?
And here’s the second answer for what the Wronskian is good for. It helps us solve ordinary differential equations. Like, particular ones. An ordinary differential equation will (normally) have several linearly independent solutions. If you know all but one of those solutions, it’s possible to calculate the Wronskian and, from that, the last of the independent solutions. Since a big chunk of mathematics — particularly for science or engineering — is solving differential equations you see why this is something valuable. Allow that it’s tedious. Tedious work we can automate, or give to research assistant to do.
One then asks what kind of differential equation would have all-but-one answer findable, and yield that last one only by long efforts of hard work. So let me show you an example ordinary differential equation:
Here , , and are some functions that depend only on the independent variable, . Don’t know what they are; don’t care. The differential equation is a lot easier of and are constants, but we don’t insist on that.
This equation has a close cousin, and one that’s easier to solve than the original. Is cousin is called a homogeneous equation:
The left-hand-side, the parts with the function that we want to find, is the same. It’s the right-hand-side that’s different, that’s a constant zero. This is what makes the new equation homogenous. This homogenous equation is easier and we can expect to find two functions, and , that solve it. If and are constant this is even easy. Even if they’re not, if you can find one solution, the Wronskian lets you generate the second.
That’s nice for the homogenous equation. But if we care about the original, inhomogenous one? The Wronskian serves us there too. Imagine that the inhomogenous solution has any solution, which we’ll call . (The ‘p’ stands for ‘particular’, as in “the solution for this particular ”.) But also has to solve that inhomogenous differential equation. It seems startling but if you work it out, it’s so. (The key is the derivative of the sum of functions is the same as the sum of the derivative of functions.) also has to solve that inhomogenous differential equation. In fact, for any constants and , it has to be that is a solution.
I’ll skip the derivation; you have Wikipedia for that. The key is that knowing these homogenous solutions, and the Wronskian, and the original , will let you find the that you really want.
My reading is that this is more useful in proving things true about differential equations, rather than particularly solving them. It takes a lot of paper and I don’t blame anyone not wanting to do it. But it’s a wonder that it works, and so well.
Don’t make your instructor so mad you have to do the Wronskian for four functions.
This and all the others in My 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be at this link. Thank you for reading.