Solving the Initial Value Problem for Impulse Response with Peak Value of 2

In summary, to find the value of k for which the response has a peak value of 2, solve the initial value problem using Laplace transforms and set y(1) = 2.
  • #1
huk
5
0
hey,

Consider the initial value problem:

y''+(gamma)y'+y=k(delta)(t-1), y(0)=0, y'(0)=0

where k is the magnitude of an impulse at t = 1 adn (gamma) is the damping coefficietnt(for resistence).
Let (gamma)=1/2. find the value of k for which the response has a peak value of 2; call this value k1..

evnthough I am taking this diff equ course, i have difficult understanding the physical meaning of this equation... what does peak value mean... and how do i find k w/ this peak value?

thanks alot
 
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  • #2
The response of a system is the solution to the initial value problem. What you want to do is solve the initial value problem via Laplace transforms...your y(t) will be a function of both time and k, set y(1) = 2 and solve for k
 
  • #3


To solve for the value of k1, we first need to understand the physical meaning of the equation. The initial value problem represents a damped harmonic oscillator with an impulse applied at t=1. The damping coefficient (gamma) represents the resistance in the system, and the impulse (k) represents the magnitude of the force applied. The peak value refers to the maximum amplitude of the response of the oscillator.

To find k1, we can use the method of undetermined coefficients. We assume a solution of the form y = Aet, where A is a constant to be determined. Substituting this into the equation, we get:

Ae^(t/2) + (gamma)Ae^(t/2) + Ae^(t/2) = k(delta)(t-1)

Simplifying, we get:

2Ae^(t/2) + (gamma)Ae^(t/2) = k(delta)(t-1)

At the peak value, the derivative of the solution is equal to 0. So we can set y' = 0 and solve for t to find the time at which the peak occurs. This gives us t = ln(2/(gamma+2)). Substituting this value into the equation, we get:

2Ae^(ln(2/(gamma+2))/2) + (gamma)Ae^(ln(2/(gamma+2))/2) = k(delta)(ln(2/(gamma+2)) - 1)

Simplifying, we get:

A(2 + (gamma)) = k(delta)(ln(2/(gamma+2)) - 1)

To find k1, we want the peak value to be 2. So we set A = 1 and solve for k:

2 + (gamma) = k(delta)(ln(2/(gamma+2)) - 1)

k = (2 + (gamma))/(delta(ln(2/(gamma+2)) - 1))

Substituting (gamma) = 1/2, we get:

k1 = (5/2)/(delta(ln(4/3) - 1))

Therefore, the value of k for which the response has a peak value of 2 is given by k1 = (5/2)/(delta(ln(4/3) - 1)). I hope this helps clarify the physical meaning of the equation and how to find the value of k for
 

Related to Solving the Initial Value Problem for Impulse Response with Peak Value of 2

1. What is the initial value problem for impulse response?

The initial value problem for impulse response is a mathematical concept that involves finding the solution to a differential equation, given an initial condition. It is used to model the behavior of a system over time, particularly in the field of engineering and physics.

2. How is the peak value of 2 determined in the initial value problem for impulse response?

The peak value of 2 is determined by the impulse response function, which is a mathematical function that describes the output of a system when it is subjected to a short, intense input. In this case, the peak value of 2 indicates the maximum response of the system to the impulse.

3. What are the applications of solving the initial value problem for impulse response with a peak value of 2?

The initial value problem for impulse response with a peak value of 2 has various applications in engineering and physics. It can be used to analyze the behavior of systems in response to sudden changes or inputs, such as in control systems, signal processing, and circuit analysis.

4. What are some methods for solving the initial value problem for impulse response with a peak value of 2?

There are several methods for solving the initial value problem for impulse response with a peak value of 2, including the Laplace transform method, the Fourier transform method, and the method of undetermined coefficients. These methods involve using mathematical techniques to find the solution to the differential equation.

5. How does the initial value problem for impulse response with a peak value of 2 relate to real-world systems?

The initial value problem for impulse response with a peak value of 2 is a useful tool for modeling and understanding the behavior of real-world systems. It allows scientists and engineers to predict how a system will respond to sudden changes or inputs, which is crucial in designing and optimizing various systems, such as electrical circuits, mechanical systems, and chemical processes.

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