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

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Discussion Overview

The discussion revolves around the historical development of the method of reduction of order in ordinary differential equations (ODEs), particularly focusing on how this method is used to find linearly independent solutions. The scope includes theoretical insights and historical perspectives on the method's formulation and application.

Discussion Character

  • Historical, Conceptual clarification, Technical explanation

Main Points Raised

  • One participant inquires about the historical origins of the method for finding linearly independent solutions to ODEs.
  • Another participant suggests that using variations of parameters to find solutions seems natural after studying differential equations, particularly in cases with repeated eigenvalues.
  • A question is raised regarding the meaning of "natural" in the context of proposing solutions.
  • Different methods of reducing the order of differential equations are acknowledged, with a request for clarification on which specific method is being discussed.
  • A participant clarifies that the method in question involves taking a known solution y1(x) and proposing that u(x) y1 is also a solution, leading to the determination of u(x).

Areas of Agreement / Disagreement

Participants express varying interpretations of the method and its historical context, indicating that multiple competing views remain regarding the specifics of the method and its naturalness.

Contextual Notes

There are unresolved aspects regarding the definitions and specific methods of reduction of order being referenced, as well as the assumptions underlying the characterization of solutions.

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