Question

transfomada de laplace laplace (t^2+1)^2

Answer 1

• Calculando a transformada de Laplace da sua função f(t), temos que ela é igual a:

$\Large\displaystyle\text{ \begin{gathered}\tt F(s)=\frac{24}{s^{5}} +\frac{4}{s^{3}} +\frac{1}{s} \end{gathered} }$

A transformada de Laplace é dada da seguinte forma:

   $\Large\displaystyle\text{ \begin{gathered}\boxed{\tt \mathcal{L} \left\{ f(t)\right\} = \int e^{-st}\cdot f(t)dt = F(s)}\end{gathered} }$

Desejamos calcular a transformada de Laplace da seguinte função:

 $\Large\displaystyle\begin{gathered}\tt \mathcal{L}\left\{(t^2+1)^2\right\} =\mathcal{L}\left\{t^4+2t^2+1\right\}= F(s)\end{gathered}$

Vamos então aplicar a seguinte propriedade:

     $\Large\displaystyle\begin{gathered}\tt \mathcal{L}\left\{ f(t)\pm g(t)\right\}= \mathcal{L}\left\{ f(t)\right\}\pm \mathcal{L}\left\{ g(t)\right\}\end{gathered}$

Logo, temos que:

$\Large\displaystyle\begin{gathered}\tt F(s)=\mathcal{L}\left\{t^4\right\}+\mathcal{L}\left\{2t^2\right\}+\mathcal{L}\left\{1\right\}\end{gathered}$

E pela lineariedade, logo:

$\Large\displaystyle\begin{gathered}\tt F(s)=\mathcal{L}\left\{t^4\right\}+2\mathcal{L}\left\{t^2\right\}+\mathcal{L}\left\{1\right\}\end{gathered}$

E essas transformadas já são tabeladas, sendo elas:

 $\Large\displaystyle\text{ \begin{gathered} \underbrace{\underline{\boxed{\tt\mathcal{L}\left\{t^n\right\} =\frac{n!}{s^{n+1}} }}}_{\Large\displaystyle\begin{gathered} \green{\tt (I)}\end{gathered}}\ \ \ \wedge\ \ \ \underbrace{\underline{\boxed{\tt\mathcal{L}\left\{1\right\}= \frac{1}{s} }}}_{\Large\displaystyle\begin{gathered} \green{\tt (II)}\end{gathered}}\end{gathered} }$

Aplicando na sua questão, temos que:

$\Large\displaystyle\begin{gathered}\tt F(s)=\mathcal{L}\left\{t^4\right\}+2\mathcal{L}\left\{t^2\right\}+\mathcal{L}\left\{1\right\}\end{gathered}$

$\Large\displaystyle\text{ \begin{gathered}\tt F(s)=\frac{4!}{s^{4+1}} +2\cdot\frac{2!}{s^{2+1}} +\frac{1}{s} \end{gathered} }$

$\Large\displaystyle\text{ \begin{gathered}\therefore \green{\underline{\boxed{\tt F(s)=\frac{24}{s^{5}} +\frac{4}{s^{3}} +\frac{1}{s}}}}\ \ \ (\checkmark). \end{gathered} }$

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