Using Fourier transformations to solve ODEs

[WhiteKnight]

Can anyone explain to me how to use FTs to solve ODEs? For example, I have:

x'' + 2x' + 4x = sin(wt) where ' is derivative w/rt time and w is the excitation frequency. I can solve this with other methods, but I'm not sure how to use FTs.

 

[Analog]

Similar to Laplace transforms. You are converting a time varying system to the frequency domain, then use simple algebra. The whole point of these transforms is to get out of doing Calc., and using algebraic skills to find the answers.

 

[Regs]

:Q

 

[WhiteKnight]

So if the FT is defined as ...exp(-i(littleomega)t) then the derivatives just become multiplication by -i(littleomega), right? i.e. x" = -(littleomega)^2 * exp(-i(littleomega)t)

 

[KLin]

Originally posted by: Regs

:Q

 

[WhiteKnight]

^

 

[StormRider]

Basically what YellowFiero said. 😉

You can avoid using complicated integrals to obtain a solution (in the time domain) to using simple algebra (multiplication) to obtain the solution (in the frequency domain).

Then you use the inverse transformation to transform the solution from the frequency domain back to the time domain. And this usually just involves "eye-balling" the solution and seeing that it resembles a pattern in a table of solutions.

 

[WhiteKnight]

Alright, thanks guys. I think I have it figured out now.