A function f(x) satisfies the differential equation \frac{{d}^{2}f}{d{x}^{2}}- {\omega }^{2}f=- \delta (x- a) where \omega is positive. The Fourier transform \hat{f}(k)=f\int_{- \infty }^{+\infty } dx{e}^{ikx}f(x) of f, and the solution of the equation are, respectively,

Fourier Transform — CSIR Net PYQ (Question 2)

A Fourier Transform previous-year question (PYQ) from Mathematical Physics (CSIR Net). Read the question, check the correct answer, then practise the full topic.

Question 2 of 6 · 5.0 marks

A function

f (x)

satisfies the differential equation

\frac{d ^{2} f}{d x ^{2}} - ω^{2} f = - δ (x - a)

where

ω

is positive. The Fourier transform

\hat{f} (k) = f \int_{- \infty}^{+ \infty} d x e^{ik x} f (x)

f

, and the solution of the equation are, respectively,

1 $\frac{e ^{ik a}}{k ^{2} + ω ^{2}}$ and $\frac{1}{2 ω} (e^{- ω ∣ x - a ∣} + e^{ω ∣ x - a ∣})$
2 $\frac{e ^{ik a}}{k ^{2} + ω ^{2}}$ and $\frac{1}{2 ω} e^{- ω ∣ x - a ∣}$
3 $\frac{e ^{ik a}}{k ^{2} - ω ^{2}}$ and $\frac{1}{2 ω} (e^{- iω ∣ x - a ∣} + e^{iω ∣ x - a ∣})$
4 $\frac{e ^{ik a}}{k ^{2} - ω ^{2}}$ and $\frac{1}{2 iω} (e^{- ω ∣ x - a ∣} - e^{iω ∣ x - a ∣})$

Correct answer: Option 2

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