Question

Question 3. (a) Consider the following ODE $\dot{y}(t) + 2y(t) = r(t)$, with $r(t) = 3 - u(t - 5)$ and initial condition $y(0) = 10$. Recall that $u(t - 5) = \begin{cases} 0 & \text{if } t < 5\\ 1 & \text{if } t \ge 5 \end{cases}$. (i) (5 marks) Find $Y(s) = \mathcal{L}(y(t))$, the Laplace transform of $y(t)$. (ii) (8 marks) Find $y(t)$ by computing the inverse Laplace transform of $Y(s)$. (iii) (2 marks) Find $\lim_{t \to \infty} y(t)$. (b) (4 marks) Find the Laplace transform of $f(t) = t \cos(3t)$. Justify your answer.

          Question 3.
(a) Consider the following ODE
$\dot{y}(t) + 2y(t) = r(t)$, 
with $r(t) = 3 - u(t - 5)$ and initial condition $y(0) = 10$. Recall that $u(t - 5) = \begin{cases} 0 & \text{if } t < 5\\ 1 & \text{if } t \ge 5 \end{cases}$.
(i) (5 marks) Find $Y(s) = \mathcal{L}(y(t))$, the Laplace transform of $y(t)$.
(ii) (8 marks) Find $y(t)$ by computing the inverse Laplace transform of $Y(s)$.
(iii) (2 marks) Find $\lim_{t \to \infty} y(t)$.
(b) (4 marks) Find the Laplace transform of $f(t) = t \cos(3t)$. Justify your answer.

Question 3.
(a) Consider the following ODE
ẏ(t) + 2y(t) = r(t),
with r(t) = 3 - u(t - 5) and initial condition y(0) = 10. Recall that u(t - 5) = 0 if t < 5
1 if t ≥ 5.
(i) (5 marks) Find Y(s) = ℒ(y(t)), the Laplace transform of y(t).
(ii) (8 marks) Find y(t) by computing the inverse Laplace transform of Y(s).
(iii) (2 marks) Find limt →∞ y(t).
(b) (4 marks) Find the Laplace transform of f(t) = t cos(3t). Justify your answer.

Added by Sergio G.

Question

Please give Ace some feedback