Question

Encontrar as transformadas de Laplace para as funções abaixo:

Answer 1

Olá Andreia!

Item a)

Sabe-se que $\mathsf{\mathcal{L} \left \{ t^n \cdot e^{at} \right \} = \frac{n!}{(s - a)^{n + 1}}}$. Isto posto,

$\\ \mathsf{\mathcal{L} \left \{ - 2t \cdot e^{- t} \right \}}= \\\\ \mathsf{- 2 \cdot \mathcal{L} \left \{t \cdot e^{- t} \right \}}= \\\\ \mathsf{- 2 \cdot \frac{1!}{(s + 1)^2}} = \\\\ \boxed{\mathsf{- \frac{2}{(s + 1)^2}}}$

Item b)

Para resolver este item e o próximo precisamos saber que $\mathsf{\mathcal{L} \left \{ t^n \right \} = \frac{n!}{s^{n + 1}}}$. Daí

$\\ \mathsf{\mathcal{L} \left \{ 5t^4 \right \} = 5 \cdot \mathsf{\mathcal{L} \left \{t^4 \right \}}} \\\\ \mathsf{\mathcal{L} \left \{ 5t^4 \right \} = 5 \cdot \frac{4!}{s^{4 + 1}}} \\\\ \boxed{\mathsf{\mathcal{L} \left \{ 5t^4 \right \} = \frac{120}{s^5}}}$

 Item c)

$\\ \mathsf{\mathcal{L} \left \{ - 3t^3 + 2t^2 - t + 5 \right \} = \mathsf{- 3 \cdot \mathcal{L} \left \{t^3 \right \} + 2 \cdot \mathcal{L} \left \{t^2 \right \} - \mathcal{L} \left \{t \right \} + 5 \cdot \mathcal{L} \left \{1 \right \}}} \\\\ \mathsf{\mathcal{L} \left \{ - 3t^3 + 2t^2 - t + 5 \right \} = \mathsf{- 3 \cdot \frac{3!}{s^{3 + 1}} + 2 \cdot \frac{2!}{s^{2 + 1}} - \frac{1!}{s^{1 + 1}} + 5 \cdot \frac{1}{s}}} \\\\ \boxed{\mathsf{\mathcal{L} \left \{ - 3t^3 + 2t^2 - t + 5 \right \} = \mathsf{- \frac{18}{s^4} + \frac{4}{s^3} - \frac{1}{s^2} + \frac{5}{s}}}}$

 Espero ter ajudado!