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## Homework Statement

Without solving the differential equation, find the differential equation that solves Fourier transformation of given differential equation for ##a>0##.

a) ##y^{'}+axy=0##

b) For what ##a## is the solution of part a) an eigenfunction of Fourier Transform

## Homework Equations

##f(x)## ....---.... ##F(f)(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi i x\xi } dx##

## The Attempt at a Solution

I have to apologize in advance. The instructions are weird even in my language, I can't imagine how horrible they sound in English.

##F(y^{'})=2\pi i \xi F(y)##

and ##F(axy)=\int _{-\infty }^{\infty }axy(x)e^{-2\pi i x\xi } dx=\frac{a}{-2\pi i}\frac{\partial }{\partial \xi}F(y)##

The differential equation is than: ##2\pi i \xi F(y)-\frac{a}{-2\pi i}\frac{\partial }{\partial \xi}F(y)=0##.

That should be the answer to part a).

part b):

Here is where I have problems. For eigenfunctions ##F(y)=y##.

##2\pi i \xi y-\frac{a}{2\pi i}\frac{\partial }{\partial \xi}y=0##

Therefore ##y=exp(-\frac{2\pi ^2}{a}\xi ^2)##.

Now this has to be wrong... So, my question? How do I find the answer for part b).