I am revisiting ODE. I understand that ODE constructs relationship between change of the dependent/output variable with the dependent variable itself plus independent/input variable. But when finding solution why do we assume a particular solution in advance with the aim to find the constants? Am I missing something?
P.S.: I am an engineering student, so it would be helpful not going into too much abstraction.
The answer is this: mathematicians go to a lot of lengths to prove existence and uniqueness of solutions of DE's under various conditions. If you can show both of those, then any method whatever that gets a solution of the DE is a valid method. Guess-and-check is one of the most valuable. It depends for its success on the previous efforts of a large number of extremely talented and intelligent people.