# Solving DE Homework: g'(t) + ag(t) = \delta(t-\xi)

• squenshl
In summary: Now you can do what you wish with it.In summary, the conversation discusses solving, sketching and checking the accuracy of a differential equation using a test function. The solution is given as g(t) = c1e-at + ea(\xi-t)H(t-\xi) where H(x) is the Heaviside function. The conversation also mentions the option of using Mathematica to plot the solution and work with a test function.
squenshl

## Homework Statement

Solve, sketch & check it's accuracy by operating on a test function of g'(t) + ag(t) = $$\delta$$(t-$$\xi$$), g(t) = 0, t < $$\xi$$

## The Attempt at a Solution

I have solved it getting g(t) = c1e-at + ea($$\xi$$-t)H(t-$$\xi$$) where H(x) is the Heaviside function. How do I sketch this & operate it on a test function.

squenshl said:

## Homework Statement

Solve, sketch & check it's accuracy by operating on a test function of g'(t) + ag(t) = $$\delta$$(t-$$\xi$$), g(t) = 0, t < $$\xi$$

## The Attempt at a Solution

I have solved it getting g(t) = c1e-at + ea($$\xi$$-t)H(t-$$\xi$$) where H(x) is the Heaviside function. How do I sketch this & operate it on a test function.

You can plot it manually if you wish. Just choose some values for a and e and c. But why not, if you're studying DEs, learn how to work with them in Mathematica. Here's the code I would use to "sketch" it and use a "test" function with the solution. Try and figure out what I'm doing if you're interested:

Code:
e = 5;
a = 1;
mysol = y /. DSolve[y'[t] + a y[t] == DiracDelta[t - e], y, t] // First
Plot[mysol[t] /. C[1] -> 1, {t, 0, 5}]

I got the sketch, so do I investigate a test function in the usual way.

squenshl said:
I got the sketch, so do I investigate a test function in the usual way.

Hi. The act of solving it numerically implicitly generates a particular (test) solution to the DE. I mean you choose some values for a, e and the arbitrary integration constant, C[1], in my code, then run the numeric integrator NDSolve. That "creates" a numeric function which satisfies the DE.

## 1. How do I solve this differential equation?

To solve this differential equation, you can use the method of variation of parameters. This involves finding a particular solution by substituting a function of t into the equation and solving for the parameters. Then, the general solution can be found by adding the particular solution to the homogeneous solution, which is found by setting a = 0 and solving the resulting equation.

## 2. What is the purpose of the delta function in this equation?

The delta function, denoted as δ(t-ξ), is a mathematical concept used to represent a point of concentration in a function. In this equation, it indicates that there is a sudden change in the function at t = ξ. It is often used in differential equations to model instantaneous events.

## 3. How do I choose the appropriate values for a and ξ in this equation?

The values of a and ξ will depend on the specific problem you are trying to solve. The value of a represents the rate of change of the function g(t) and ξ represents the point at which the delta function occurs. These values can be determined from the given information about the system or by applying physical principles to the problem.

## 4. Can I use a different method to solve this equation?

Yes, there are other methods that can be used to solve this type of differential equation. Some common methods include the Laplace transform method, the method of undetermined coefficients, and the method of integrating factors. However, the method of variation of parameters is often the most straightforward approach for solving this equation.

## 5. Are there any real-world applications of this type of differential equation?

Yes, this type of differential equation has various applications in fields such as physics, engineering, and economics. It can be used to model systems with sudden changes or impulses, such as a sudden force acting on a moving object or a sudden change in market demand. It can also be used to study the behavior of electrical circuits and mechanical systems.

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