Question 34 on signals from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
Q34. Consider the differential equation
}{dt}+2\frac{dy(t)}{dt}+y(t) = \delta(t)})
with |}_{t=0^{-}}=-2)
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