Solve Impulse-Diffy eq. Homework: y''+y=δ(t-2π)cos(t), y(0)=0, y'(0)=1

• polarbears
In summary, the conversation discusses a differential equation with a delta function and cosine. The problem is approached using Laplace transforms, and the integral involving the delta function is discussed. The topic of the conversation is identified as differential equations and Laplace transforms.
polarbears

Homework Statement

$$y''+y=\delta (t-2\pi )cos(t)$$
$$y(0)=0,y'(0)=1$$

The Attempt at a Solution

The left side is (s^2+1)Y(s)-1=RHS

My problem is the fact that cosine is being multiplied by the delta function. I put it in the form of an intergral but I don't know what to do from there.

Well, good! Delta functions usually make integrals trivial. What integral did you get?

hmm, i don't know what is this question, may i know, what topic should i study for this question?

Differential Equations -Laplace transforms

OHHHH waitttt does the delta function just determine the bound of my intergral?

1. What is an impulse-diffy equation?

An impulse-diffy equation is a type of differential equation that includes an impulse function, also known as a Dirac delta function. This type of equation is commonly used in physics and engineering to model sudden, short-lived changes in a system.

2. How do I solve an impulse-diffy equation?

To solve an impulse-diffy equation, you will need to use a combination of integration techniques and algebraic manipulation. In some cases, you may also need to apply boundary conditions to find the specific solution for the given initial conditions.

3. What do the given initial conditions mean in the problem?

The given initial conditions, y(0)=0 and y'(0)=1, represent the initial position and velocity of the system at time t=0. These conditions help to determine the unique solution to the equation.

4. How do I deal with the delta function in this equation?

The delta function, δ(t-2π), represents a spike or impulse occurring at time t=2π. To solve the equation, you will need to use the properties of the delta function, such as the sifting property, to integrate over the impulse and incorporate it into the solution.

5. What is the solution to the given impulse-diffy equation?

The solution to the given impulse-diffy equation, y''+y=δ(t-2π)cos(t), y(0)=0, y'(0)=1, is y(t)=sin(t) for t≠2π and y(2π)=1. This solution takes into account the given initial conditions and the impulse function at t=2π.

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