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.