# Method of reduction of order

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 $xy_1(x)$. It is a natural step to generalize this thinking when your coefficients aren't constant, by proposing a solution of the form $a(x)y_1(x)$ and see what does the function a must fulfill in order to span a solution.

What do you mean by natural?

HallsofIvy