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## Main Question or Discussion Point

How did people come up with this method of finding lin independent solutions to ODEs?

- Thread starter pivoxa15
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How did people come up with this method of finding lin independent solutions to ODEs?

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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.

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What do you mean by natural?

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HallsofIvy

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