Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**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 \( 0 \leq t \leq 4 \).](https://content.bartleby.com/qna-images/question/2a4cecdd-94f1-45ae-b6d2-43e39cc00a4c/171a9896-d2c2-4d61-a7c3-38be6b264046/hzhysvs_processed.png)
Transcribed Image Text:**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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