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

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SUMMARY

This discussion focuses on solving non-homogeneous ordinary differential equations (ODEs) with coupled equations using Fourier series. The process involves first determining the general solution of the homogeneous equation, followed by finding particular solutions for each of the 25 in-homogeneous equations derived from the Fourier series terms. The participants confirm that while manual calculations are labor-intensive, software tools like MATLAB can efficiently handle the computations, allowing for a more streamlined approach to solving the equations without needing to solve all 25 individually.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with Fourier series and their applications
  • Proficiency in MATLAB for numerical computations
  • Knowledge of linear algebra concepts related to coupled equations
NEXT STEPS
  • Learn how to implement Fourier series in MATLAB for solving ODEs
  • Study the method of undetermined coefficients for particular solutions of ODEs
  • Explore numerical methods for solving coupled differential equations
  • Investigate the use of symbolic computation in MATLAB for ODEs
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Mathematicians, engineers, and students specializing in differential equations, particularly those interested in numerical solutions and applications of Fourier series in solving coupled ODEs.

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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