PDE fouriertransform

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Im supposed to find the solution of ODE (10) with $\lim_{y \to \infty} u(x,y) = 0$. I have done the same as the soultionsheet and got the general solution (11). But the next step confuses me, howcome we knpw thath the D(w) = 0 when we use the foruiertransform? Can somebody please explain the theory behind the calculation?

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