When solving differential equations, it seems like we often have to guess the form of the solution beforehand. How does one know what to try? It seems natural to the authors of textbooks to try say a power series solution or some other form, but I don't quite understand how they know. Also, there tends to also be the method of guessing a form of the solution then verifying later that it is valid, but won't we risk unwittingly compromise generality? Thanks.
Guessing the form of solutions when solving differential equations
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ordinary-differential-equations