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Sagot :
The given function s/(s^2 +3s -4) is proved with the help of inverse Laplace theorem.
According to the statement
we have to find the inverse of the Laplace theorem with the help pf the given theorem in the statement.
So, For this purpose, we know that the
Laplace transformation is a transformation of a function f(x) into the function g(t) that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.
Now, We assume you want to find the inverse transform of [tex]s/(s^2 +3s -4).[/tex]
This can be written in partial fraction form as
[tex]\frac{(4/5)}{(s+4)} + \frac{(1/5)}{(s-1)}[/tex]
which can be found in a table of transforms to be the transform of
[tex]\frac{4}{5} e^{-4t} + \frac{1}{5} e^t[/tex]
There are a number of ways to determine the partial fractions. They all start with factoring the denominator.
[tex]s^2 +3x -4 = (s+4)(s-1)[/tex]
After that, you can postulate the final form and determine the values of the coefficients that make it so.
For example:
[tex]\frac{A}{(s+4)} + \frac{B}{s-1} = (A+B)s + \frac{(4B-A)}{(s^2 +3x -4)}[/tex]
This gives rise to two equations:
(A+B) = 1
(4B-A) = 0.
So, The given function s/(s^2 +3s -4) is proved with the help of inverse Laplace theorem.
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Question:
Use appropriate algebra and theorem 7.2.1 to find the given inverse laplace transform. (write your answer as a function of t.) ℒ−1 s s2 + 3s − 4.
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