Answer
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Step 1:: Graph the forcing function, $f(t)$
Graph of $f(t)$:
f(t) = \begin{cases} 1 & \text{if } t \geq \pi, \ 0 & \text{if } t < \pi. \end{cases} t / f(t) --------- 0 / 0 \ pi / 1 \ 3pi / 1
Step 2:: Find the Laplace transform of the differential equation
Taking the Laplace transform of both sides, we get: \mathcal{L}\{\bar{y}+ 4\dot{y}+ 5y\} = \mathcal{L}\{f(t)\} Using linearity property and known Laplace transforms, we have: \mathcal{L}\{\bar{y}\} + 4\mathcal{L}\{\dot{y}\} + 5\mathcal{L}\{y\} = \mathcal{L}\{1\} - \mathcal{L}\{u(t-\pi)\}
Step 3:: Apply initial conditions
where $Y(s)$ is the Laplace transform of $y(t)$.
Using these initial conditions, we can find the Laplace transforms: \begin{align*} \mathcal{L}\{\bar{y}\} &= sY(s) - y(0) = sY(s), \ \mathcal{L}\{\dot{y}\} &= sY(s) - y(0) = sY(s), \end{align*}
Step 4:: Solve for $Y(s)$
Substituting the Laplace transforms from Step 3 into the equation from Step 2, we get: Simplifying, we have:
Step 5:: Partial fraction decomposition
The first term can be decomposed as: So,
Step 6:: Inverse Laplace transform
\begin{align*} \end{align*}
Step 7:: Simplified expressions for $y(t)$
y(2$) = 1 - u(2$)
Step 8:: Sketch the graph of the solution
The graph of the solution is a step function that increases by $\frac{1}{9}$ at $t = \pi$.
t / y(2$) --------- 0 / 0 \ pi / 0 \ 3pi / 1 / 9
Final Answer
- Simplified expressions for $y(t)$: see Step 7 - Sketch of the graph: see Step 8
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