Find the Laplace transform of:
f(t)=t-2e^(3t)
Just consult your table
L{t} = 1/s^2
L{e^3t} = 1/(s-3)
so, L{t - 2e^(3t)} = 1/s^2 - 2/(s-3)
To find the Laplace transform of f(t), we will use the linearity property of Laplace transforms and find the transforms of each term separately.
1. Transform of t:
The Laplace transform of t is given by:
L{t} = 1/(s^2)
2. Transform of 2e^(3t):
To find the transform of 2e^(3t), we need to use the shifting property of Laplace transforms. The shifting property states that for a function h(t) multiplied by e^(at), the transform is given by:
L{e^(at) * h(t)} = H(s - a)
In this case, a = 3 and h(t) = 2. Therefore, the transform of 2e^(3t) is given by:
L{2e^(3t)} = 2 / (s - 3)
Using the linearity property, we can combine the two transforms:
L{f(t)} = L{t} - L{2e^(3t)}
= 1/(s^2) - 2 / (s - 3)
Therefore, the Laplace transform of f(t) is 1/(s^2) - 2 / (s - 3).
To find the Laplace transform of the function f(t) = t - 2e^(3t), we can break it down into two separate Laplace transforms: one for t and one for -2e^(3t).
The Laplace transform of t is denoted by L{t}, and it can be found using the formula for the Laplace transform of t^n where n is a positive integer. In this case, since t is a first-degree polynomial with respect to t, we can use the formula:
L{t} = 1/(s^2), where s is the complex variable.
So, the Laplace transform of t is 1/(s^2).
Next, let's find the Laplace transform of -2e^(3t). The general formula for the Laplace transform of e^(at) is:
L{e^(at)} = 1/(s-a), where s is the complex variable and a is a constant.
Using this formula, the Laplace transform of -2e^(3t) becomes:
L{-2e^(3t)} = -2/(s-3).
Finally, we can combine the individual Laplace transforms to find the Laplace transform of f(t):
L{f(t)} = L{t} + L{-2e^(3t)} = 1/(s^2) - 2/(s-3).
Therefore, the Laplace transform of f(t) = t - 2e^(3t) is 1/(s^2) - 2/(s-3).