Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. y" — 6y' = 8(t — 5), a. Find the Laplace transform of the solution. Y(s) = L{y(t)} = b. Obtain the solution y(t). y(t) = y(0) = 5, y'(0) = 0. c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 5. y(t) = { if 0 < t < 5, if 5 < t <∞.
Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. y" — 6y' = 8(t — 5), a. Find the Laplace transform of the solution. Y(s) = L{y(t)} = b. Obtain the solution y(t). y(t) = y(0) = 5, y'(0) = 0. c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 5. y(t) = { if 0 < t < 5, if 5 < t <∞.
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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![Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function.
\[ y'' - 6y' = \delta(t - 5), \quad y(0) = 5, \quad y'(0) = 0. \]
a. Find the Laplace transform of the solution.
\[ Y(s) = \mathcal{L}\{y(t)\} = \]
[Input box for solution]
b. Obtain the solution \( y(t) \).
\[ y(t) = \]
[Input box for solution]
c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at \( t = 5 \).
\[ y(t) = \begin{cases}
\text{\tt[Input box for solution]} & \text{if } 0 \leq t < 5, \\
\text{\tt[Input box for solution]} & \text{if } 5 \leq t < \infty.
\end{cases} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1069a960-ab44-4829-baef-0330e3e99a25%2Fa662cead-3404-4465-a8c3-815954d01ef4%2F20ere7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function.
\[ y'' - 6y' = \delta(t - 5), \quad y(0) = 5, \quad y'(0) = 0. \]
a. Find the Laplace transform of the solution.
\[ Y(s) = \mathcal{L}\{y(t)\} = \]
[Input box for solution]
b. Obtain the solution \( y(t) \).
\[ y(t) = \]
[Input box for solution]
c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at \( t = 5 \).
\[ y(t) = \begin{cases}
\text{\tt[Input box for solution]} & \text{if } 0 \leq t < 5, \\
\text{\tt[Input box for solution]} & \text{if } 5 \leq t < \infty.
\end{cases} \]
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