Partial Differential Equations — CSIR Net PYQ (Question 1)

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

Question 1 of 2 · 5.0 marks

Let

f (x, t)

be a solution of the wave equation

\frac{\partial ^{2} f}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} f}{\partial x ^{2}}

in 1 -dimension. If at

t = 0, f (x, 0) = e^{- x^{2}}

and

\frac{\partial f}{\partial t} (x, 0) = 0

for all

x

, then

f (x, t)

for all future times

t > 0

is described by

1 $e^{- (x^{2} - v^{2} t^{2})}$
2 $e^{- (x - v t)^{2}}$
3 $\frac{1}{4} e^{- (x - v t)^{2}} + \frac{3}{4} e^{- (x + v t)^{2}}$
4 $\frac{1}{2} [e^{- (x - v t)^{2}} + e^{- (x + v t)^{2}}]$

Correct answer: Option 4

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