**IVP Problem Overview****Problem Statement:**Consider the following first-order initial value problem (IVP):\[\frac{dy}{dt} + \frac{1}{2}y = f(t), \quad y(0) = 0.\]where $ f(t) $ is given as the piecewise function:\[f(t) = \begin{cases} 4, & 0 \leq t < 2 \\0, & 2 \leq t < \infty \end{cases}\]Find the (unique) solution, $ y(t) $, to this IVP, evaluate $ y(4) $, and sketch the graph of $ y(t) $ on the interval $ 0 \leq t \leq 4 $.**Steps to Solve:****S1:** Solve the initial value problem on the interval $ 0 \leq t \leq 2 $.**S2:** Evaluate $ y(2) $ (label this value as $ y_2 $).**S3:** Solve the IVP, $ L[y] = f(t) $ for $ t > 2 $ with initial condition $ y(2) = y_2 $.**S4:** Express your solution in piecewise form and sketch its graph on $ 0 \leq t \leq 4 $.

Answered: Image /qna-images/answer/171a9896-d2c2-4d61-a7c3-38be6b264046.jpg

Answered: 4. Consider the following first-order…

4. Consider the following first-order initial value problem (IVP), dy +zy = f(t), 1 y(0) = 0. dt where f(t) is given as the piecewise function: 4, 0 2 with initial condition y(2) = Y2.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**IVP Problem Overview**

**Problem Statement:**

Consider the following first-order initial value problem (IVP):

\[
\frac{dy}{dt} + \frac{1}{2}y = f(t), \quad y(0) = 0.
\]

where $ f(t) $ is given as the piecewise function:

\[
f(t) =
\begin{cases}
4, & 0 \leq t < 2 \\
0, & 2 \leq t < \infty
\end{cases}
\]

Find the (unique) solution, $ y(t) $, to this IVP, evaluate $ y(4) $, and sketch the graph of $ y(t) $ on the interval $ 0 \leq t \leq 4 $.

**Steps to Solve:**

**S1:** Solve the initial value problem on the interval $ 0 \leq t \leq 2 $.

**S2:** Evaluate $ y(2) $ (label this value as $ y_2 $).

**S3:** Solve the IVP, $ L[y] = f(t) $ for $ t > 2 $ with initial condition $ y(2) = y_2 $.

**S4:** Express your solution in piecewise form and sketch its graph on $ 0 \leq t \leq 4 $.

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