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

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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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Thread 'Volterra equation of the second kind'

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Thread 'Visual Representation of Separation of Varables'

Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
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