QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
3. Use Laplace transforms to solve the initial value problem $\bar{y}+ 4 \dot{y}+ 5 y=f(t), y(0)= 0, \dot{y}(0)= 0$ , where $f(t)= 1 -u(t-\pi)$. As the first step, graph the forcing function, $f(t)$. Once you have solved the equation, please write simplified expressions for $y$ when $0 \leq t<\Pi$ and when $t \geq \boldsymbol{\Pi}$ Also, please sketch the graph of the solution or attach a desmos-generated graph. For full credit, please show all steps of the process. Don't hesitate to add your personal commentary.
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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)\}
Final Answer
- Simplified expressions for $y(t)$: see Step 7 - Sketch of the graph: see Step 8
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