please help show the step on how to get the correct asnwer. i have attached the correct answer Question: solve the given initial-value problem.sketch the graph of both the forcing function and the solution. y″ + y′ +3y = u(t –2) y(0)= 0 y′(0) = 1
please help show the step on how to get the correct asnwer. i have attached the correct answer Question: solve the given initial-value problem.sketch the graph of both the forcing function and the solution. y″ + y′ +3y = u(t –2) y(0)= 0 y′(0) = 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please help show the step on how to get the correct asnwer. i have attached the correct answer
Question: solve the given initial-value problem.sketch the graph
of both the forcing function and the solution.
y″ + y′ +3y = u(t –2)
y(0)= 0
y′(0) = 1
![The image contains a mathematical expression composed of two main parts:
1. **First Expression:**
\[
\frac{1}{3} \, u(t - 2) \left( \left( 1 - e^{-3(t - 2)\sqrt{2}/2} \cos \frac{\sqrt{3}}{2} (t - 2) - 2 \sqrt{3} \, e^{-3(t - 2)\sqrt{2}/2} \sin \frac{\sqrt{3}}{2} (t - 2) \right) + \frac{2}{\sqrt{3}} \, e^{-3\sqrt{2}} \sin \frac{\sqrt{3}}{2} \right)
\]
- Here, \( u(t - 2) \) denotes a unit step function that shifts the function in time by 2 units.
- The expression includes exponential decay factors and a combination of cosine and sine functions, which might indicate a damped oscillatory behavior.
2. **Second Expression:**
\[
\frac{5}{2} \left( 1 - e^{-t}(\cos 2t + \sin 2t) - u(t - \pi)[1 - e^{-(t-\pi)}(\cos 2t + \sin 2t)] \right)
\]
- This part also includes exponential decay and trigonometric functions.
- The term \( u(t - \pi) \) indicates a unit step function with a shift of \(\pi\).
- The expression models another type of damped oscillation with shifts in time.
Overall, the expressions seem to represent solutions to differential equations involving damped oscillatory systems, possibly relevant in physics or engineering contexts. They involve time-shifted unit step functions, exponential terms, and trigonometric components, suggesting their application in advanced signal processing or control systems. There are no graphs or diagrams present in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe31b6b9e-ba6c-4dca-9489-09cb6373ae58%2Fdb6ce188-1ec6-4ceb-b2f3-85df88592180%2Fietiqh_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression composed of two main parts:
1. **First Expression:**
\[
\frac{1}{3} \, u(t - 2) \left( \left( 1 - e^{-3(t - 2)\sqrt{2}/2} \cos \frac{\sqrt{3}}{2} (t - 2) - 2 \sqrt{3} \, e^{-3(t - 2)\sqrt{2}/2} \sin \frac{\sqrt{3}}{2} (t - 2) \right) + \frac{2}{\sqrt{3}} \, e^{-3\sqrt{2}} \sin \frac{\sqrt{3}}{2} \right)
\]
- Here, \( u(t - 2) \) denotes a unit step function that shifts the function in time by 2 units.
- The expression includes exponential decay factors and a combination of cosine and sine functions, which might indicate a damped oscillatory behavior.
2. **Second Expression:**
\[
\frac{5}{2} \left( 1 - e^{-t}(\cos 2t + \sin 2t) - u(t - \pi)[1 - e^{-(t-\pi)}(\cos 2t + \sin 2t)] \right)
\]
- This part also includes exponential decay and trigonometric functions.
- The term \( u(t - \pi) \) indicates a unit step function with a shift of \(\pi\).
- The expression models another type of damped oscillation with shifts in time.
Overall, the expressions seem to represent solutions to differential equations involving damped oscillatory systems, possibly relevant in physics or engineering contexts. They involve time-shifted unit step functions, exponential terms, and trigonometric components, suggesting their application in advanced signal processing or control systems. There are no graphs or diagrams present in the image.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
