Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function: x"- 8x' 8(t-3), x(0) 1, x'(0) = 0. In the following parts, use h(te) for the Heaviside function he(t) if necessary. a. Find the Laplace transform of the solution. L {a(t)}(s) = b. Obtain the solution a(t). x(t) = c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 3. x(t) = 8. 8. if 0 < t < 3, if 3 < t < 0.
Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function: x"- 8x' 8(t-3), x(0) 1, x'(0) = 0. In the following parts, use h(te) for the Heaviside function he(t) if necessary. a. Find the Laplace transform of the solution. L {a(t)}(s) = b. Obtain the solution a(t). x(t) = c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 3. x(t) = 8. 8. if 0 < t < 3, if 3 < t < 0.
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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Question
Oo.122.
Subject :- Advance Mathematic
![Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function:
x" - 8x' 8(t-3), x(0) = 1, x'(0) = 0.
In the following parts, use h(t - c) for the Heaviside function he(t) if necessary.
a. Find the Laplace transform of the solution.
L {x(t)}(s) =
b. Obtain the solution (t).
x(t) =
c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 3.
x(t) =
9-
if 0 < t < 3,
if 3 < t <∞0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a79f5fd-2bdd-484b-9c0d-a83d48b55247%2Fe8d4907f-dfb9-40b8-9339-beed63c3fcbe%2Fig0vdhg_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:
x" - 8x' 8(t-3), x(0) = 1, x'(0) = 0.
In the following parts, use h(t - c) for the Heaviside function he(t) if necessary.
a. Find the Laplace transform of the solution.
L {x(t)}(s) =
b. Obtain the solution (t).
x(t) =
c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 3.
x(t) =
9-
if 0 < t < 3,
if 3 < t <∞0.
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