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d²y(1) dy d₁² dt where y(0) = 0; y'(0) = 1. +8y= u(1)

d²y(1) dy d₁² dt where y(0) = 0; y'(0) = 1. +8y= u(1)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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The image shows a differential equation commonly found in control systems and dynamic analysis studies. The equation is given by: \[ \frac{d^2 y(t)}{dt^2} + 6 \frac{dy}{dt} + 8y = u(t) \] This is a second-order linear differential equation, where: - \( y(t) \) is the dependent variable, usually representing a system's response over time. - \( u(t) \) is an input function or external force applied to the system. - The coefficients of the derivative terms are constants: 6 for the first derivative and 8 for the function itself. The initial conditions provided are: - \( y(0) = 0 \), meaning the initial value of the function is zero. - \( y'(0) = 1 \), indicating the initial rate of change of the function is one. This setup can be used to analyze how the system reacts over time, considering the impact of the input \( u(t) \).
Transcribed Image Text:The image shows a differential equation commonly found in control systems and dynamic analysis studies. The equation is given by: \[ \frac{d^2 y(t)}{dt^2} + 6 \frac{dy}{dt} + 8y = u(t) \] This is a second-order linear differential equation, where: - \( y(t) \) is the dependent variable, usually representing a system's response over time. - \( u(t) \) is an input function or external force applied to the system. - The coefficients of the derivative terms are constants: 6 for the first derivative and 8 for the function itself. The initial conditions provided are: - \( y(0) = 0 \), meaning the initial value of the function is zero. - \( y'(0) = 1 \), indicating the initial rate of change of the function is one. This setup can be used to analyze how the system reacts over time, considering the impact of the input \( u(t) \).
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