Question 34 on signals from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
Q34. Consider the differential equation
The numerical value of
Let us Laplace transform to find and later
, where with real numbers and .
Using integration by parts,
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,
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
 GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf
 Wiki entry on Laplace transform of function’s derivative
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