Let me resume the Fourier thing.

Back and forth transform pair are:

,

.

Since I was concerning about solving ODE using Fourier transform in a similar manner as Lapalce’s, I want to see what the Fourier transform of derivatives (with respect to ) of look like. For the 1st derivative, I get:

.

by using integration by parts. If vanishes at infinity (), then the 1st term becomes 0, so I get:

Now higher derivatives are easy. Fourier transform of the 2nd derivative, for example, is:

.

Okay. Let me pick a 2nd order ODE:

.

Taking Fourier transform of both sides, I get:

,

.

Finally I reach by inverse-transforming both sides:

.

mmm… but is the integral on the right hand side looks so clumsy… It is a double-integral actually.

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