31:T40d,md`## The Wronskian That much is all well and good. Normally one would be obliged to proceed in a more logical order (e.g., to explain how to actually "solve" the equation above), but it's perhaps better to jump to a rather non-trivial analogy, and it has to do with the Wronskian. If you are like me, you probably have forgotten it. The [Wronskian](https://en.wikipedia.org/wiki/Wronskian) of two (linearly independent) solutions ${tex`u_1, u_2`} of the equation ${tex`af'' + bf' + cf = 0`} is defined to be ${tex.block`W = \begin{vmatrix} u_1 & u_2 \\ u_1' & u_2' \end{vmatrix} = u_1 u_2' - u_1' u_2`} (the determinant of the so-called *fundamental matrix*). It is so defined that, if we replace ${tex`u_1`} and ${tex`u_2`} by another basis of solutions, ${tex`W`} would only change by a constant scalar. So, at least for us, the Wronskian (up to a constant) is associated with the differential equation itself, independent of the choice of ${tex`u_1`} and ${tex`u_2`}. [Check that the Wronskian of ${tex`f''+f=0`} is a constant.] `32:T470,md`What's remarkable about the Wronskian is that it can be computed without knowing either of the solutions! It is given by Abel's formula: ${tex.block` W = W_0 \exp \int -\frac{b}{a} dt`} [Derive this by first computing ${tex`W'`}...] This gives us a recipe to find the second solution if we somehow know the first one.

Example: ${tex.block`f'' -2f'+f=0`} has ${tex`e^t`} as a solution, so that [...click to show more]

${tex.block`W = \begin{vmatrix} e^t & u_2 \\ e^t & u_2' \end{vmatrix} = e^{2t}`} which means that ${tex`u_2`} satisfies a first-order differential equation, namely ${tex.block`e^t u_2' - e^t u_2 = e^{2t}`} You might be tempted to divide ${tex`e^t`}, but instead we will divide ${tex`e^{2t}`} throughout: ${tex.block`e^{-t} u_2' - e^{-t} u_2 = 1`} and observe that the left hand side is ${tex`(e^{-t}u_2)'`}. Thus we have ${tex`u_2=te^t`} as our second solution.

Here's a departure from parallels with equations in algebra: you always have a second solution, never a "repeated root" scenario. `33:T410,md` | | algebraic equation | (linear) differential equation | |----|-------------|-------------| |prototype (n=2) | ${tex`ax^2+bx+c=0`} | ${tex`af''+bf'+cf=h`} | |solutions are | (complex) **numbers** | (meromorphic) **functions** | |coefficients are | integers ${tex`\mathbb Z`}
rational numbers ${tex`\mathbb Q`} | constants ${tex`\mathbb C`}
polynomials ${tex`\mathbb C[x]`}
rational functions ${tex`\mathbb C(x)`} | |solvable means | expressible in radicals | expressible in integrals | |(in general)
not solvable for | ${tex`n\geq 5`} | ${tex`n\geq 2`} ☹️
(variable coefficients) | |can eliminate
2nd highest term by
a change of variables | "completing the square":
${tex`y=x+\frac{b}{na}`} | "integrating factor":
${tex`g=f \cdot \exp{\int \frac{b}{na}\,dt}`} | | symmetry of the solutions | Galois group
(permuting the ${tex`n`} roots) | Lie group
(of ${tex`n\times n`} matrices) | |iterative
approximations | Newton's method | Picard iteration
(via Peano-Baker series) | `34:T4c5,md`## Nonlinear differential equations It must be stressed that all the above is *only* the linear case, in which ${tex`f, f', f''`} only appear "linearly". In some sense this is just the simplest case, and the fun begins when you have nonlinear terms in the equation. As an example, the simple pendulum with damping (air resistence that's proportional to velocity) is described by ${tex.block` \theta'' + \theta' + \sin(\theta) = 0`} Even though we can't write it down, the general solution still involves two independent parameters (two degrees of freedom), which may be determined by initial conditions: initial position ${tex`\theta(0)`} and initial velocity ${tex`\theta'(0)`}. This is formalized as the existence and uniqueness theorem for ordinary differential equations, for a rather general class of equations, but care must to taken to account for this example (h/t Allen Knutson): ${tex.block`(f')^2 + f^2 = 1`} Both ${tex`\sin`} and ${tex`\cos`} clearly solve the equation, but there are many others: ${tex`f`} rises up to ${tex`1`} like ${tex`\sin`}, and remains constant there for a short while before falling down, and stays at ${tex`-1`} for a long while, and so on. These solutions are *not* analytic. `35:[] 2f:{"title":"Differential equations, a revisionist take","cells":[{"id":0,"value":"md`# Differential equations, a revisionist take\n\nI recently came across Gian-Carlo Rota's excellent and witty piece\n\n* Gian-Carlo Rota, [Ten lessons I wish I had learned before I started teaching differential equations](https://web.williams.edu/Mathematics/lg5/Rota.pdf) (1997)\n\nKnowing that universities today still follow the same outdated curriculum, I can't help but want to give a shot at revamping it, outside the formal classroom. (Ok, part of it is also me wanting to learn about what Rota referred to as *differential algebra*.) This will read more like a blog post, until I get myself to make some interactives in JavaScript. Comments and suggestions are welcome!`","pinned":false,"hidden":false,"mode":"ojs"},{"id":582,"value":"md`> *Help on JavaScript is greatly appreciated! One thing I'd like to include is the phase diagram of the simple pendulum, with side-by-side physics simulation as in [here](https://www.myphysicslab.com/pendulum/pendulum-en.html).*`","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"md`## Differential vs. algebraic\n\nIt is obvious that the most natural way to introduce differential equations for the first time, as say in a first-year calculus course (after integration and before sequences and series), is to draw the analogy with equations in algebra. Everyone has learned about the quadratic equation:\n${tex.block`ax^2+bx+c = 0`}\nHere, ${tex`a`}, ${tex`b`}, ${tex`c`} are given, and ${tex`x`} is the unknown that we are trying to solve. You may or may not remember the *quadratic formula*:\n${tex.block` x = \\frac{-b\\pm\\sqrt{b^2 - 4ac}}{2a}`}\nThe exact formula is unimportant for us, except to note the ${tex`\\pm`} sign: there are *two* solutions, or (very rarely) one solution, and other times zero. [You are free to think about complex numbers, or to ignore them in the following.]`","pinned":false,"hidden":false,"mode":"ojs"},{"id":39,"value":"md`Now, a differential equation looks like this:\n${tex.block`af'' + bf' + cf = h`}\nThe ${tex`a`}, ${tex`b`}, ${tex`c`}, and ${tex`h`} are again given, and we are looking for *functions* ${tex`f`} that make the equality hold. The example to keep in mind is\n${tex.block`f'' + f = 0`}\nto which every student of calculus can come up with two solutions: ${tex`f(x)=\\sin x`} and ${tex`f(x)=\\cos x`}. Are there any others?\n\n... (depending on what your response is), we eventually arrive at one or another form of the *general solution*:\n${tex.block`\\begin{aligned} f(x) &= A\\cos x + B\\sin x \\\\\n&= C\\cos(x-\\phi) \\\\\n&= D e^{ix} + E e^{-ix} \\\\\n&= F e^{i(x-\\phi)}\\end{aligned}`}\n[Q: For given values of ${tex`A`} and ${tex`B`}, can you convert the solution to the other forms? There is a minor puzzling issue with the constants being complex numbers; see if you can sort it out.]`","pinned":false,"hidden":false,"mode":"ojs"},{"id":111,"value":"md`That is, while the quadratic equation has *two* solutions, a second-order differential equation should have a *two-parameter family* of solutions. Here, some basic concepts in linear algebra help to express it more clearly: the set of all solutions (or the *solution space*) is a two-dimensional vector space, of which ${tex`\\cos x`} and ${tex`\\sin x`} form a *basis* (i.e., they are *linearly independent* and any solution is a *linear combination* of them).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1290,"value":"md`This, I believe, is a basic fact about differential equations that often doesn't get across in a first-year calculus course, which sadly only deals with the (less interesting) first-order equations.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":113,"value":"md`By the way, the reason that the functions ${tex`\\cos x`}, ${tex`\\sin x`}, and ${tex`e^x`} are so important (and, along with their inverses, are deemed *elementary*) can be traced to the fact that they are solutions to the simplest differential equations ubiquitous in physics and other sciences. Another, more \"theoretical\", reason: integrals of rational functions (and those that involve a \"quadratic radical\") are nicely captured by these elementary functions. Of course the latter is also a type of differential equations, namely ${tex`f'=h`} for a known function ${tex`h`}.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":115,"value":"md`If the ${tex`a`}, ${tex`b`}, ${tex`c`} above are functions as opposed to just constants, the problem of solving the differential equation becomes considerably more difficult. And, as Rota lamented, despite the numerous \"classical\" methods and contrived examples (standard materials for an ODE course), it rarely leads to explicit solutions in terms of elementary functions, which is \"demoralizing\" to the students. Notwithstanding, two things we can be assured of: 1) the set of solutions would still look like a two-dimensional plane—that's the defining feature of a *linear* differential equation; and 2) each and every solution can (easily) be expressed as a power series, thus *analytic*, provided that the ${tex`a`}, ${tex`b`}, ${tex`c`} are analytic to begin with. More on that later.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":289,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":336,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":540,"value":"md`The Wronskian bears a striking resemblance to the sum of the two roots ${tex`r_1, r_2`} of the equation ${tex`ax^2 + bx + c = 0`}, which can be computed without knowing either root:\n${tex.block`r_1 + r_2 = -\\frac{b}{a}`}\nSame expression down to the sign! This is (the first) [Vieta's formula](https://en.wikipedia.org/wiki/Vieta%27s_formulas) [and it is very simple; you don't need the quadratic formula for it.]`","pinned":false,"hidden":false,"mode":"ojs"},{"id":444,"value":"md`Both the Wronskian and Vieta's formula generalize straightforwardly to higher-order equations, providing a recipe for deriving the ${tex`n`}-th solution if we somehow know the first ${tex`n-1`} solutions—even though the differential case may be utterly impractical, if not downright impossible.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":396,"value":"md`Now it begs the obvious question: what is the analogue of the other Vieta's formulas? For quadratic, the product of the two roots is given by\n${tex.block`r_1 r_2 = \\frac{c}{a}`}\nAnd what about the discriminant ${tex`\\Delta = b^2-4ac`}? [I do not know the answers.]`","pinned":false,"hidden":false,"mode":"ojs"},{"id":126,"value":"md`## More analogies\n\nThe various techniques for solving a (linear) differential equation with variable coefficients are akin to the tricks/formulas for the cubic and quartic equations of the Renaissance Italians, which have long fallen out of school curriculum (for good): You do this and do that, until you get something like ${tex`(\\cdots)'=h`} or ${tex`(\\cdots)^n = d`}, which you can integrate or extract the ${tex`n`}-th roots directly. This and other observations lead us to the following table of analogies:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":602,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":1106,"value":"md`The inability (or impossibility) to solve many equations in both cases is no cause for concern. We used to say that we are expanding or relaxing the notion of \"solving\" an equation; now from a modern machine-computable point of view, integration and extracting the ${tex`n`}-th roots are themselves iterative processes, hence are no different from making succesive approximations on the equation directly.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1304,"value":"md`Historically, Galois and his contemporaries thought of everything as complex numbers, which is much too vast to do algebra but serves as a convenient \"working field\" since we can easily visualize a two-dimensional plane. The vogue of today is to work with much smaller fields—finite (or at most algebraic) extensions of the field of rational numbers—*in the abstract*; it may have different \"embeddings\" in the complex plane, but everything we care about must be independent of that. \n\nThe case for differential equations is similar but with important differences: the solutions would all be meromorphic functions (on some extended domain, if necessary), but that field is actually way too vast for any particular equation. It makes more sense to start with polynomial and rational functions, and only build up small extensions when necessary (such as viewing ${tex`z^{\\lambda}`}, ${tex`\\lambda\\in\\mathbb C\\setminus \\mathbb Z`}, as an abstract entity that satisfies the equation ${tex`zf'=\\lambda f`}).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1205,"value":"md`It is of course well known that functions and numbers share many things in common; after all, we speak of equations and solutions in both cases. The standard introduction to ring theory (part of modern/abstract algebra) is essentially a unified treatment of the algebraic properties (e.g., unique factorization) of the integers and polynomials over a field.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":619,"value":"md`This is, admittedly, a somewhat unconventional (but better, IMO) way to think about differential equations: they are a blackbox that, in theory at least, \"generate\" complicated functions as solutions when you feed in simple functions as coefficients. The conventional story is to speak about *general* functions from the start, and the coefficients are often taken to be just continuous (or [Lipschitz](https://en.wikipedia.org/wiki/Lipschitz_continuity)) functions on an interval. I'm becoming more discontent with this modern approach in calculus and analysis in general, but that's for another time. [It can be argued that the natural setting for linear differential equations with polynomial coefficients is the space of *tempered* or *Schwartz distributions*; I'm more willing to accept the delta function as a function than an arbitrary mapping from ${tex`\\mathbb R`} to ${tex`\\mathbb R`}.]\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1402,"value":"md`## Resultant?\n\nTo follow up on the Wronskian, there is another quantity associated with two polynomials that decides if they have a root in common. It is known as the resultant, and likewise has two very different expressions. Denoting the two polynomials by ${tex`p`} and ${tex`q`}, and the roots by ${tex`r_i`} and ${tex`s_j`}, respectively, then the resultant of ${tex`p`} and ${tex`q`}, on the one hand, is given by\n${tex.block`R(p,q) = \\prod_{i,j} (r_i - s_j)`}\nwhile on the other hand, it is equal to a certain determinant of order ${tex`\\deg(p)+\\deg(q)`}, with entries given by the coefficients of ${tex`p`} and ${tex`q`}.\n\nIt would be interesting to have an analogue for differential equations.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":501,"value":"md`### Further readings\n\nDifferential algebra, as a distinct subject, is mostly modeled on Galois theory (for algebraic equations). For a brief introduction, see\n*\tJohn H. Hubbard, Benjamin E. Lundell, A First Look at Differential Algebra, *The American Mathematical Monthly*, **118**(3): 245-261 (2011)\n\nIn addition, there is a rich theory of differential equations over the complex domain developed over the course of the 19th century, culminating in another connection with group theory. This led to the theory of automorphic forms, a central theme of number theory today, though its origin in differential equations is rarely taught. For a detailed and technical history, see\n\n* Jeremy J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré (2008)\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":629,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":1160,"value":"md`\nIn some sense, many if not most nonlinear systems (i.e., systems of two or more equations that are *coupled*) are **chaotic**, dashing our hope to make long-term predictions with differential equations. See the [double pendulum](https://beta.observablehq.com/@travisdoesmath/double-pendulums-are-chaotic) for a classic example. (It might be argued that space and time are fundamentally discrete, and differential equations are ultimately an approximation for the physical world; but some very simple system like the [logistic map](https://observablehq.com/@gcalmettes/routes-to-chaos) also exhibits complicated behavior.) With that said, it must be rather remarkable that there are **integrable** systems (e.g, the KdV equation) that exhibit the opposite phenomenon, with \"rigid\" long-term *exact* solutions.`","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W35","stdlib":"1","titling":"auto"}