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Find the inverse Laplace Transforms of the following functions: A. F(s) = \frac{s + 1}{s(s^2 +...

Question:

Find the inverse Laplace Transforms of the following functions:

A. {eq}F(s) = \frac{s + 1}{s(s^2 + 2s + 2)} {/eq}

B. {eq}F(s) = \frac{-2x}{(x^2 + 4)^2} {/eq}

C. {eq}F(s) = \frac{e^{-7s}}{s^2 + 2s + 2} {/eq}

Inverse Laplace Transform:

Some important formulae for ILT are

{eq}\begin{align} & {{\mathcal{L}}^{-1}}\left( \frac{1}{s} \right)=1 \\ & {{\mathcal{L}}^{-1}}\left( \frac{1}{s-a} \right)={{e}^{at}} \\ & {{\mathcal{L}}^{-1}}\left( \frac{a}{{{s}^{2}}+{{a}^{2}}} \right)=\sin \left( at \right) \\ & {{\mathcal{L}}^{-1}}\left( \frac{s}{{{s}^{2}}+{{a}^{2}}} \right)=\cos \left( at \right) \\ & {{\mathcal{L}}^{-1}}\left( \frac{{{e}^{-cs}}}{s} \right)={{u}_{c}}\left( t \right) \\ \end{align} {/eq}

Answer and Explanation: 1

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Part (A):

{eq}F\left( s \right)=\frac{s+1}{s\left( {{s}^{2}}+2s+2 \right)} {/eq}

Perform partial fraction decomposition

{eq}\begin{align} &...

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First-Order Linear Differential Equations

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Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.


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