Perhaps delta function or inverse Laplace transform?

In summary, the conversation is about how to solve a differential equation involving a Dirac delta function and Laplace transforms. The participants give advice on how to approach the problem, including using integration by parts and the convolution theorem for Laplace transforms. The goal is to find the solution for i(t).
  • #1
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0
Hello everyone,
i have this question and not even sure how to approach it:

[tex]\frac {di}{dt}+4i+3\int_{0^-}^t{i(z)dz = 12(t-1)u(t-1) [/tex]

and [tex]i(0^-) = 0[/tex]

find [tex] i(t) [/tex]

last topics we covered were laplace transforms (and inverse) and dirac delta function.
At least some hint to get me started would be a great help.

EDIT:
oh, and again u(t) = 1 for t >= 0 and u(t) = 0 elsewhere.
 
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  • #2
I would get rid of the integral by taking the derivative of the entire equation - which gives you a second order differential equation and leads you into the LaPlace transform.
 
  • #3
Laplace transform the equation directly. To find the transform of the integral, just do an integration by parts. That will avoid complications on the right side. :)
 
  • #4
thanks for replies, as i looked further through the book, we actually have an entry in the table for this integral, but what do I do with [tex]i[/tex] for Laplace transform? it does not have u(t)...
 
  • #5
Ev,

I presume that your goal is to solve for i(t). After performing the Laplace transforms, you will have an algebraic equation for I(s). The right side will have two terms. One will relate to the initial value of i and the other will be a product of an algebraic quantity with the Laplace transform of the integral containing u. You should have no difficulty inverting the first part and you should be able to do something with the second using the convolution theorem for Laplace transforms.

Let us know what you end up with!
 

1. What is the delta function and how is it used in mathematics and science?

The delta function, also known as the Dirac delta function, is a mathematical concept that represents an infinitely small and infinitely tall spike at a specific point on a graph. It is often used in physics and engineering to model point sources of energy or mass. In mathematics, it is used to define and solve differential equations and in probability to represent a probability distribution at a single point.

2. How is the delta function related to the inverse Laplace transform?

The inverse Laplace transform is a mathematical operation that takes a function in the complex frequency domain and transforms it back into the time domain. The delta function is often used as a test function or as an initial condition for a differential equation in the Laplace domain, and the inverse Laplace transform is used to solve for the corresponding function in the time domain.

3. Can the delta function be represented as a continuous function?

No, the delta function is not a continuous function. It is a distribution or generalized function that is defined at a single point and has infinite magnitude at that point. However, it can be approximated by a sequence of continuous functions that become narrower and taller as the sequence approaches infinity.

4. What is the relationship between the delta function and the Kronecker delta?

The Kronecker delta is a discrete function that takes the value of 1 when the input variables are equal and 0 otherwise. It can be seen as a discrete version of the delta function, where instead of being infinitely small and infinitely tall, it takes discrete values of 1 or 0 at specific points. The delta function and the Kronecker delta are often used interchangeably in different fields of mathematics, depending on the context.

5. Are there any real-world applications of the delta function or inverse Laplace transform?

Yes, there are many real-world applications of the delta function and inverse Laplace transform. In physics, they are used to model and solve problems in fields such as electromagnetism, quantum mechanics, and fluid dynamics. In engineering, they are used to analyze and design systems in areas such as control theory, signal processing, and circuit analysis. In economics, they are used to model and analyze markets and financial systems. These are just a few examples of the many applications of these mathematical concepts in various fields.

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