Handbook of Exact Solutions for Ordinary Differential Equations, Second Edition

Rating:
Summary: Analytic solutions are still preferable
Review: An amazing tome. 6200 ordinary differential equations that are exactly solvable. Typically, when you study ODEs, your text will describe a standard pedagogical set, like systems of DEs with constant coefficients, etc. Look at Boyce and diPrima's "Elementary Differential Equations" as a good example of this.

But just as people have compiled tables of numerical values of useful functions, like Abramowitz and Stegun's "Handbook of Mathematical Functions", did anyone do likewise for ODEs? Well, Polyanin et al have done so.

These days, many who face solving an ODE might resort to doing so numerically, since computers and software packages have become so powerful. But analytic solutions are still always preferable, assuming that they exist and you can find them. The reasons are threefold. Firstly, they are more compact to encode than tables, from a computational viewpoint. Secondly, it is often easier to search for understanding in a known functional form than in a table. Thirdly, if you have a solution in the form of a function, and that function is differentiable or integrable, then you may be able to gain more insight, or apply the answer to broader usage, by doing so.

Thus analytic solutions are desirable. The problem is in finding them. That is where this book has value. It encapsulates a lot of specialised ODE solving techniques, applied to reams of equations.

As an aside, the book also shows a qualitative difference between Russia and the US. The Soviets always lagged behind the US in computing. So by necessity, Soviet scientists emphasised more the traditional pencil and paper approach to solving equations. Whereas Americans were more likely to toss it over to a nearby computer. This has lead in the US to the deprecating of courses and texts in differential equation solving. Hence if you are an American researcher facing an ODE with no obvious solution, it may be quite difficult for you, ab initio, to find a solution. So check this book first for answers.