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

In summary, the method of finding linearly independent solutions to ODEs involves using variation of parameters, which is a common procedure in differential equations. When working with constant coefficients, it is natural to propose a solution in the form xy_1(x), but this can be generalized to non-constant coefficients by proposing a solution in the form a(x)y_1(x) and finding the necessary conditions for a. This method is also known as reducing the order of a differential equation.
  • #1
pivoxa15
2,255
1
How did people come up with this method of finding lin independent solutions to ODEs?
 
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  • #2
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.
 
  • #3
What do you mean by natural?
 
  • #4
There are a number of different ways of reducing the order of differential equation, depending on the differential equation. Which one do you mean?
 
  • #5
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))
 

What is the "method of reduction of order"?

The method of reduction of order is a mathematical technique used to solve second-order linear differential equations by reducing them to first-order equations. It is based on the assumption that one solution of the original equation is known, and uses this solution to find a second linearly independent solution.

When is the method of reduction of order used?

The method of reduction of order is used when solving second-order linear differential equations with variable coefficients. It is particularly useful when one solution of the equation is known, but the general solution cannot be easily determined using other methods.

How does the method of reduction of order work?

The method of reduction of order works by assuming that one solution of the original equation is known, and then using this solution to construct a new equation. This new equation can then be solved to find a second linearly independent solution, which can be used to form the general solution of the original equation.

What are the steps involved in using the method of reduction of order?

The steps involved in using the method of reduction of order are:

  • Assume one solution of the original equation is known
  • Construct a new equation by substituting this solution into the original equation
  • Solve the new equation to find the second linearly independent solution
  • Use the two solutions to form the general solution of the original equation

What are the advantages of using the method of reduction of order?

The method of reduction of order has several advantages, including:

  • It can be used to solve a wide range of second-order linear differential equations
  • It is particularly useful when one solution of the equation is known
  • It can reduce the complexity of the problem by reducing a second-order equation to a first-order equation
  • It is a straightforward and systematic method that can be easily applied to different equations

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