Non-Homogeneous ODEs with Coupled Equations: Solving with Fourier Series?

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Discussion Overview

The discussion revolves around solving non-homogeneous ordinary differential equations (ODEs) that involve forcing functions represented in terms of Fourier series. Participants explore methods for finding solutions, particularly in the context of coupled equations and the feasibility of using computational tools like MATLAB.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant suggests starting with the general solution of the homogeneous equation before finding particular solutions for the in-homogeneous equations, noting that there are 25 such equations due to the linearity of the problem.
  • Another participant questions the practicality of solving these equations by hand and inquires about the possibility of using MATLAB for computation.
  • A reply emphasizes that it is unnecessary to solve all 25 equations individually, proposing that one can solve a representative set instead, specifically mentioning solving for 'n' without specific values.
  • A later post introduces a new scenario involving two second-order simultaneous non-homogeneous equations and asks if there is a method to solve them, indicating a shift in focus to coupled equations.

Areas of Agreement / Disagreement

Participants generally agree on the approach of solving the homogeneous part first and then addressing the in-homogeneous components, but there is no consensus on the best method for handling the complexity of the equations or the use of computational tools.

Contextual Notes

The discussion does not resolve the specific methods for solving the coupled equations or the effectiveness of MATLAB in this context. There are assumptions about the linearity and the nature of the equations that remain unexamined.

Who May Find This Useful

This discussion may be useful for students and practitioners dealing with non-homogeneous ODEs, particularly in the context of Fourier series and coupled equations, as well as those interested in computational methods for solving differential equations.

jason.bourne
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how do we solve an ODE which has forcing function in terms of Fourier series?
i have attached a pdf file of the problem.
 

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At first you should find the general solution of the homogenous equation.Then you should find the particular solutions of the in-homogenous equations. I use plural words because you have in fact 25 in-homogenous equations with the driving functions being the terms in the Fourier series. That's because the equation is linear and so you can just consider each term the only one which is there and find the particular solution corresponding only to that term and then add the particular solutions together and to the general solution of the homogenous equation to get the answer.
 
it will be very laborious right by hand calculation? is it possible to solve on MATLAB by writing code?
 
jason.bourne said:
it will be very laborious right by hand calculation? is it possible to solve on MATLAB by writing code?

You're not going to actually solve 25 differential equations!
Just solve it with n,without giving it specific values,Which means you're going to solve only 2 differential equations one of which is the representative of 24 differential equations.
But yes,you can solve it with softwares like MatLab too.
 
yeah. got it. thanks for helping me Shyan. yes i realized it was silly thing to ask.

Shyan, let say if we have 2 second order simultaneous non homogeneous equations which are coupled, is there any way to solve it?
i have included a typical problem in an attachment.
 

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