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 graphof both the forcing function and the solution.y″ + y′ +3y = u(t –2)y(0)= 0y′(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.

Here the fircing function isu(t−2)={0t<21t≥2Given differential equationy″+y′+3y=u(t−2)The initial…

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

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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.

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