d²x dx +2 + 5x = f(t) dt² dt By hand show the work to solve for X(s). Let L[x(t)] = X(s) & L[ƒ (t)] = F(s). Plot the solution/response up to time t = 4 seconds, for the case where f(t) = a.) a step function of height 2 b.) a ramp function with slope 0.5 c.) an impulse of magnitude 1 d.) a sine wave with amplitude 2 and frequency 1
d²x dx +2 + 5x = f(t) dt² dt By hand show the work to solve for X(s). Let L[x(t)] = X(s) & L[ƒ (t)] = F(s). Plot the solution/response up to time t = 4 seconds, for the case where f(t) = a.) a step function of height 2 b.) a ramp function with slope 0.5 c.) an impulse of magnitude 1 d.) a sine wave with amplitude 2 and frequency 1
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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![The image contains the following mathematical problem for an educational purpose:
**Problem Statement:**
Solve the following differential equation by hand to find \( X(s) \):
\[
\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 5x = f(t)
\]
Use the Laplace transform, where \( \mathcal{L}[x(t)] = X(s) \) and \( \mathcal{L}[f(t)] = F(s) \).
Plot the solution/response up to time \( t = 4 \) seconds for different types of functions \( f(t) \) as follows:
a. A step function of height 2
b. A ramp function with slope 0.5
c. An impulse of magnitude 1
d. A sine wave with amplitude 2 and frequency 1
**Instructions:**
1. Solve the differential equation using Laplace transforms.
2. Plot the responses for each type of input function over the specified time period.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff46cd712-1d53-471c-8e4d-5e78c3c80932%2F8ea4e975-476c-41d8-b903-6be8746453b6%2Ffknki9j_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains the following mathematical problem for an educational purpose:
**Problem Statement:**
Solve the following differential equation by hand to find \( X(s) \):
\[
\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 5x = f(t)
\]
Use the Laplace transform, where \( \mathcal{L}[x(t)] = X(s) \) and \( \mathcal{L}[f(t)] = F(s) \).
Plot the solution/response up to time \( t = 4 \) seconds for different types of functions \( f(t) \) as follows:
a. A step function of height 2
b. A ramp function with slope 0.5
c. An impulse of magnitude 1
d. A sine wave with amplitude 2 and frequency 1
**Instructions:**
1. Solve the differential equation using Laplace transforms.
2. Plot the responses for each type of input function over the specified time period.
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