The Laplace transform of y(t) = e^t cos(t) is:
(A) \frac{4(s-1)}{[(s-1)^2+4]^2}
(B) \frac{2(s+1)}{[(s+1)^2+4]^2}
(C) \frac{4(s+1)}{[(s+1)^2+4]^2}
(D) \frac{2(s-1)}{[(s-1)^2+4]^2}
Answer (1)
The problem here is to find the Laplace transform of the function y ( t ) = e t cos ( t ) . To solve this, it's useful to recall a relevant formula from Laplace transform theory.
The Laplace transform of a function of the form e a t ⋅ cos ( b t ) is given by:
L { e a t ⋅ cos ( b t )} = ( s − a ) 2 + b 2 s − a
In the function y ( t ) = e t ⋅ cos ( t ) , we have a = 1 and b = 1 . Let's substitute these values into the formula:
L { e t ⋅ cos ( t )} = ( s − 1 ) 2 + 1 2 s − 1
Simplifying further, this becomes:
L { e t ⋅ cos ( t )} = ( s − 1 ) 2 + 1 s − 1
Given the options, none directly matches this standard form. However, analyzing each choice for their correctness partially or as slight derivations could include further transforms such as linear modification:
Option (A) is [( s − 1 ) 2 + 4 ] 2 4 ( s − 1 )
Option (B) is [( s + 1 ) 2 + 4 ] 2 2 ( s + 1 )
Option (C) is [( s + 1 ) 2 + 4 ] 2 4 ( s + 1 )
Option (D) is [( s − 1 ) 2 + 4 ] 2 2 ( s − 1 )
None matches the transform precisely. But, each transform may have special considerations. The simplest direct answer based on known logic is not amongst these; however, carefully analyzing compositions could offer insight when alternates blend linearly in complex equations involving additional transformations.
Hence, based on simplified standard formula knowledge, none select exactly what straightforward methodology suggests if further modification found proposition holds potential outcomes, standard N o n e .
