Discovering the Method of Reduction of Order in ODEs: A Historical Perspective

[pivoxa15]

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

 

[AiRAVATA]

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.

 

[pivoxa15]

What do you mean by natural?

 

[HallsofIvy]

There are a number of different ways of reducing the order of differential equation, depending on the differential equation. Which one do you mean?

 

[Pseudo Statistic]

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))