Single approach to solving differential equations

Question

I am an engineer who uses mathematics for applications. I have learnt how to solve differential equations, both ordinary and partial. My impression has been that solving differential equations is all about knowing a bag of diverse tricks: separation of variables, reduction in order, power series method, etc.

I would like to know if there is a single approach that would work for differential equations. I don't mind if the approach is tedious or if it involves successive approximations. All I wish for is that the procedure of solving differential equations be mechanical in nature, and applicable to widest possible variety of differential equations. I first thought that writing unknown function as Taylor series and successively finding the unknown coefficients is a very general, although tedious (which is alright with me), approach to solving differential equations. However I later learnt that it works only if the expansion is carried about a regular point, otherwise it gives nonsensical answer.

Recently I have begun studying one-parameter group theoretic method for solving differential equations, and the author of a book promises it is a very general method. I wished to ask your opinion regarding this and whether there are any other general approaches which could be learnt with minimum prerequisites. Thanks in advance for any advice.

Answer 1

As Zach Boyd said here, Lie's approach "does unify a lot of existing methods". For a good overview, see:

• Eric J. Albright et al., “Symmetry Analysis of Differential Equations: A Primer” (Los Alamos National Lab. (LANL), Los Alamos, NM (United States), May 29, 2018).

Olver writes in the introduction to his Applications of Lie Groups to Differential Equations:

When beginning students first encounter ordinary differential equations they are, more often than not, presented with a bewildering variety of special techniques designed to solve certain particular, seemingly unrelated types of equations, such as separable, homogeneous or exact equations. Indeed, this was the state of the art around the middle of the nineteenth century, when Sophus Lie made the profound and far-reaching discovery that these special methods were, in fact, all special cases of a general integration procedure based on the invariance of the differential equation under a continuous group of symmetries.

Similarly, Ibragimov writes in the preface to his A Practical Course in Differential Equations and Mathematical Modeling:

over 400 types of integrable second-order ordinary differential equations were accumulated due to ad hoc approaches and summarized in voluminous catalogues [e.g., EqWorld]. …Lie group analysis reduces the classical 400 types of equations to 4 types only!

See also the beginning of the first paper by Sophus Lie in Lie Group Analysis: Classical Heritage, in which he gives a good historical overview of all the methods of solving differential equations; he begins (p. 4):

In my opinion, the major part of papers on differential equations published within the last 120 years can be divided into four or five categories having much in common.

Answer 2

It turns out that this question has been asked in one form or another by many people through the years, and it's complicated.

First, it depends on what is meant by solving the equation. Differential equations can describe a vast range of phenomena, from turbulent flow to crystal growth to dynamic plasticity. The "closed form" solutions that can be written down explicitly turn out to be inadequate to describe all of that.

A natural next step is to look for series solutions, but as you noted, many equations develop irregularities, for instance shock waves, which cannot really be described with series easily. People have tried things like shock tracking that handle these singularities separately, but it is hard.

Another approach is using Lie groups, which you have alluded to. This does unify a lot of existing methods, but it is still essentially limited to situations where a tractable closed form solution is available.

The most common modern approach to the problem it to not expect a closed form or series approximation in general ( although this is sometimes possible and useful) but instead look for either useful properties of the solution (e.g. existence, bounds on derivatives, etc.) or try to evaluate the solution at different points via numerical simulation. Another perspective on this technique is that numerical discretizations provide the sequential approximation you are looking for.

A bit disappointing, but that is the state of things. Lie group methods are cool though. Definitely study them :)

Answer 3

Solving differential equations is very much about improvisation. And that does require a bag of tricks. It's much more than just plugging things into formulas.Sometimes you arrive at solutions through what feels like guesswork. It's very much a nonlinear though process rather than the linear, step-by-step logic based approach you use for writing proofs.