Question 34 on signals from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

## Q34. Consider the differential equation

## with and

## The numerical value of

## is

## (A) -2

## (B) -1

## (C) 0

## (D) 1

## Solution

Let us Laplace transform to find and later

The Laplace transform of function’s derivative is

, where with real numbers and .

Using integration by parts,

.

Rearranging,

.

Extending this to find the Laplace Transform of the second derivative of the function,

.

Coming back to the problem,

Taking Laplace transform,

.

To find the inverse Laplace transform, let us revisit the Laplace transform for some simple functions.

For , the Laplace transform is,

.

From the discussion in the post on Q11 in GATE 2012,

.

Also from the earlier discussion in this post,

Applying the above equations to find the inverse Laplace transform

.

Taking the differential,

.

Plugging in ,

**Based on the above, the right choice is (D) 1**

** **

## References

[1] GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf

[2] Wiki entry on Laplace transform of function’s derivative

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