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1. Homework Statement
I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice:
2. Homework Equations
a) The Fourier Transform
b) The Inverse Fourier Transform
3. The Attempt at a Solution
I started by applying the Fourier Transform to the equation, this gives me:
[tex]\mathrm{i} k \hat{f}(k)  2\hat{f}(k) = \hat{\phi}(k)[/tex]
I then solve for the transformed function:
[tex]\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k2}[/tex]
At this point I want to invert this function in order to find the solution to the ODE:
[tex]f(x) = F^{1}\left( \frac{\hat{\phi}(k)}{\mathrm{i}k2} \right) = F^{1}\left( \hat{\phi}(k) \right) F^{1}\left( \frac{1}{\mathrm{i}k2} \right) = \phi(x) F^{1}\left( \frac{1}{\mathrm{i}k2} \right) [/tex]
This is where I'm a bit unsure of how to proceed. As I said in the beginning I'm quite inexperienced with this. So I am unsure if I have made any mistakes or wrong assumptions up till this point. How do I proceed with the inversion of the function?
I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice:
2. Homework Equations
a) The Fourier Transform
b) The Inverse Fourier Transform
3. The Attempt at a Solution
I started by applying the Fourier Transform to the equation, this gives me:
[tex]\mathrm{i} k \hat{f}(k)  2\hat{f}(k) = \hat{\phi}(k)[/tex]
I then solve for the transformed function:
[tex]\hat{f}(k) = \frac{\hat{\phi}(k)}{\mathrm{i}k2}[/tex]
At this point I want to invert this function in order to find the solution to the ODE:
[tex]f(x) = F^{1}\left( \frac{\hat{\phi}(k)}{\mathrm{i}k2} \right) = F^{1}\left( \hat{\phi}(k) \right) F^{1}\left( \frac{1}{\mathrm{i}k2} \right) = \phi(x) F^{1}\left( \frac{1}{\mathrm{i}k2} \right) [/tex]
This is where I'm a bit unsure of how to proceed. As I said in the beginning I'm quite inexperienced with this. So I am unsure if I have made any mistakes or wrong assumptions up till this point. How do I proceed with the inversion of the function?
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