How did people come up with this method of finding lin independent solutions to ODEs?
You mean historically?
Its a common procedure in differential equations to use some sort of variation of parameters in order to find solutions. If you ask me, after studying this mathematical objects for a while, it seems quite natural to suggest such a solution.
When you have a second order ode with constant coefficients, if the characteristic equation has repeated eigenvalues (resonance), then you are one solution short, so you propose a solution in the form [itex]xy_1(x)[/itex]. It is a natural step to generalize this thinking when your coefficients aren't constant, by proposing a solution of the form [itex]a(x)y_1(x)[/itex] and see what does the function a must fulfill in order to span a solution.
What do you mean by natural?
There are a number of different ways of reducing the order of differential equation, depending on the differential equation. Which one do you mean?
From what I understand, he's talking about the method where, if y1(x) is a solution to an ODE, then u(x) y1 is also a solution. (And you go on to find u(x))
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