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.
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 :)