How Can Discontinuous Driving Functions Be Solved Without LaPlace Transforms?

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In summary, the conversation discusses the use of LaPlace Transforms to solve differential equations with discontinuous driving functions, and whether it is possible to solve these equations using other methods such as Undetermined Coefficients or Variation of Parameters. The idea of solving each continuous differential equation separately and adjusting the constants to match the graphs is mentioned, but it is noted that this method is tedious. The connection between the characteristic polynomial and the Laplace transform is also brought up, with the observation that they are the same for homogeneous linear ODEs. The conversation ends with a question about how to use other methods to solve these equations and a lack of understanding about the role of the characteristic polynomial in LaPlace Transforms.
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Skrew
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I have been studying LaPlace Transforms and I have learned how they are used to solve DE's with discontinuous driving functions, which is certainly interesting but I was wondering is it possible to solve the same DE's using other methods such as Undetermined Coefficients or Variation of Parameters(and how would it be done)?

I have an idea of how you might be able to so, solving each continuous DE separably then adjusting the constants so the graphs meet consecutively but I don't know if this is correct.
 
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Yes it is. but it is very tedious.

As a side note, the characteristic polynomial of a homogenous linear ODE is the same as the denominator of the Laplace transform. This is no coincidence.
 
  • #3
Dickfore said:
Yes it is. but it is very tedious.

As a side note, the characteristic polynomial of a homogenous linear ODE is the same as the denominator of the Laplace transform. This is no coincidence.

So how would you go about doing it?

I was also wondering about why the characteristic polynomial shows up, to be completely honest I don't have a great understanding of LaPlace Transforms beyond them being a transform and something that is used to solve linear DE IVP's.
 

FAQ: How Can Discontinuous Driving Functions Be Solved Without LaPlace Transforms?

What is a discontinuous driving function?

A discontinuous driving function is a type of mathematical function that has abrupt changes, or discontinuities, in its value. This means that the function is not continuous, or smooth, and has distinct intervals where it changes.

What are some examples of discontinuous driving functions?

Some examples of discontinuous driving functions include step functions, impulse functions, and Heaviside functions. These functions all have sudden changes in their value, making them discontinuous.

How are discontinuous driving functions used in science?

Discontinuous driving functions are commonly used in physics and engineering to model real-world systems that have abrupt changes, such as electrical circuits and mechanical systems. They can also be used in mathematical modeling to approximate more complex functions.

What are the challenges in working with discontinuous driving functions?

One of the main challenges in working with discontinuous driving functions is that they are not differentiable at the points of discontinuity. This means that traditional calculus techniques cannot be applied, and special methods, such as distribution theory, must be used.

Can discontinuous driving functions have practical applications?

Yes, discontinuous driving functions have many practical applications in science and engineering. They can be used to model real-world systems and make predictions about their behavior. They are also useful in signal processing and control systems, among other areas.

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