Short Answer

Determine the inverse Laplace transform of the given function.
$\frac{2 s + 16}{s^{2} + 4 s + 13}$ .

The inverse laplace transform of the given function is $2 e^{- 2 t} \cos 3 t + 4 e^{- 2 t} \sin 3 t$ .

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Determining the inverse laplace transform

For a given transfer function H, the Inverse Laplace Transform takes the output Y(s) and determines what X(s) it is in terms of (s).
Consider a function F(s), if there is a function f(t) that is continuous on $[0, \infty)$ and satisfies $L {f} = F$ then we say that f(t) is the inverse Laplace transform of F(s) and employ the notation
$f = L^{- 1} {F}$
$L^{- 1} \{\frac{n!}{{(s - a)}^{n + 1}}\} = e^{a t} t^{n}, n = 1, 2, \dots$

Find inverse laplace transform for the given function

The given function is $\frac{2 s + 16}{s^{2} + 4 s + 13}$ .

Simplify $\frac{2 s + 16}{s^{2} + 4 s + 13}$ as follows:

$\begin{array}{rcl} \frac{2 s + 16}{s^{2} + 4 s + 4 + 9} & = & \frac{2 (s + 8)}{(s^{2} + 4 s + 4) + 9} \\ = & \frac{2 (s + 2) + 12}{{(s + 2)}^{2} + {(3)}^{2}} \\ = & \frac{2 (s + 2)}{{(s + 2)}^{2} + {(3)}^{2}} + \frac{12}{{(s + 2)}^{2} + {(3)}^{2}} \end{array}$

Find the inverse laplace transform of $\frac{2 s + 16}{s^{2} + 4 s + 4 + 9} = \frac{2 (s + 2)}{{(s + 2)}^{2} + {(3)}^{2}} + \frac{12}{{(s + 2)}^{2} + {(3)}^{2}}$ using $L^{- 1} \{\frac{b}{{(s - a)}^{2} + {(b)}^{2}}\} = e^{a t} \sin b t$ and $L^{- 1} \{\frac{b}{{(s - a)}^{2} + {(b)}^{2}}\} = e^{a t} \cos b t$ as:

$\begin{array}{rcl} L^{- 1} \{\frac{2 s + 16}{s^{2} + 4 s + 4 + 9}\} & = & L^{- 1} \{\frac{2 (s + 2)}{{(s + 2)}^{2} + {(3)}^{2}} + \frac{12}{{(s + 2)}^{2} + {(3)}^{2}}\} \\ = & 2 L^{- 1} \{\frac{s + 2}{{(s + 2)}^{2} + {(3)}^{2}}\} + 4 L^{- 1} \{\frac{3}{{(s + 2)}^{2} + {(3)}^{2}}\} \\ = & 2 e^{- 2 t} \cos 3 t + 4 e^{- 2 t} \sin 3 t \end{array}$

Therefore, the inverse laplace transform of the given function is $2 e^{- 2 t} \cos 3 t + 4 e^{- 2 t} \sin 3 t$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Determine the inverse Laplace transform of the given function.
$\frac{2 s + 16}{s^{2} + 4 s + 13}$ .

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Determining the inverse laplace transform

Find inverse laplace transform for the given function

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Determine the inverse Laplace transform of the given function.2s+16s2+4s+13.

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Determining the inverse laplace transform

Find inverse laplace transform for the given function

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Determine the inverse Laplace transform of the given function.
$\frac{2 s + 16}{s^{2} + 4 s + 13}$ .