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

In summary, to solve an ODE with a forcing function in terms of Fourier series, you must first find the general solution of the homogenous equation and then solve for the particular solutions of the in-homogenous equations. This can be done by considering each term in the Fourier series as its own equation and adding the particular solutions together with the general solution of the homogenous equation. While this process may be laborious by hand, it can also be solved using software such as MATLAB. Additionally, for simultaneous non-homogeneous equations, it is possible to solve them by treating them as 2 differential equations and using software to solve them.
  • #1
jason.bourne
82
1
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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  • problem.pdf
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  • #2
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.
 
  • #3
it will be very laborious right by hand calculation? is it possible to solve on MATLAB by writing code?
 
  • #4
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.
 
  • #5
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.
 

Attachments

  • New Doc 16.pdf
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1. What is a non-homogeneous ODE?

A non-homogeneous ODE (ordinary differential equation) is a type of differential equation where the dependent variable and its derivatives appear alongside an independent variable, and the equation is not equal to zero. This is in contrast to a homogeneous ODE, where the equation is equal to zero.

2. How do you solve a non-homogeneous ODE?

To solve a non-homogeneous ODE, you can use one of several methods such as the method of undetermined coefficients, variation of parameters, or Laplace transforms. These methods involve finding a particular solution, which is added to the general solution of the corresponding homogeneous ODE to obtain the complete solution.

3. What is the difference between a non-homogeneous and homogeneous ODE?

The main difference between a non-homogeneous and homogeneous ODE is that the former has a non-zero term, while the latter has a zero term. This means that the general solution of a homogeneous ODE will only contain a linear combination of solutions, while the general solution of a non-homogeneous ODE will also include a particular solution.

4. Can a non-homogeneous ODE have multiple solutions?

Yes, a non-homogeneous ODE can have multiple solutions. This is because it is possible to have different particular solutions that can be added to the general solution of the corresponding homogeneous ODE. However, the general solution of a non-homogeneous ODE will always have the same number of arbitrary constants as the general solution of its corresponding homogeneous ODE.

5. What are some real-world applications of solving non-homogeneous ODEs?

Solving non-homogeneous ODEs is important in many fields of science and engineering, such as physics, chemistry, biology, and economics. It can be used to model and predict a wide range of natural phenomena, including population growth, heat transfer, chemical reactions, and economic growth.

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