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Find the Laplace inverse of the following functions:
(a) F(s)=1+s

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Final answer:

The inverse Laplace transform of the function F(s) = 1 + s is delta(t) + t, combining the delta function and the time variable t.

Step-by-step explanation:

The student has asked to find the Laplace inverse of the function F(s) = 1 + s.

We need to recall that the Laplace transform of the function f(t) = 1 is 1/s, and the Laplace transform of f(t) = t is 1/s2. Since Laplace transforms are linear, we can find the inverse Laplace transform of F(s) by considering the inverse Laplace transform of each term separately.

The inverse Laplace transform of 1 is a delta function, which is usually excluded when looking for a real function as the inverse Laplace. For the term s, its inverse Laplace transform is the derivative of the delta function, but in practice for real-valued functions, we refer to the transform of the unit step function. Therefore the inverse Laplace of s is t itself.

Combining the two, the inverse Laplace transform of F(s) = 1 + s is delta(t) + t, where delta(t) is the delta function.

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