Laplace transform of an ODE with a non-smooth forcing function

Laplace transform. He mentions that the origin and singularities/discontinuities at the origin need to be considered. Additionally, he points out that the right-hand side of the problem is a delta function. In summary, Jason is seeking guidance on how to handle the discontinuity at the origin when using the Laplace transform to solve a differential equation involving the ramp function and a delta function.
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Homework Statement
How do I deal with non-smooth forcing functions if I want to solve the Laplace transform of an ode.
Relevant Equations
See below
Suppose I'm solving
$$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?
 
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Check the screenshot I attached:
Screen Shot 2020-06-21 at 7.33.38 AM.png
 
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  • #3
How is your course defining the Laplace transform? There is more than one way, especially with regards to how you treat the origin and singularities/discontinuities at the origin such as the one you are dealing with. By the way, the right-hand side of your problem is a delta functino: ##x^{\prime\prime}(t) = \delta(t)##.

Jason
 
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1. What is a Laplace transform?

A Laplace transform is a mathematical tool used to convert a function of time into a function of complex frequency. It is often used in engineering and physics to solve differential equations and analyze systems.

2. What is an ODE?

An ODE (ordinary differential equation) is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model physical phenomena in fields such as physics, engineering, and economics.

3. What does it mean for a forcing function to be non-smooth?

A non-smooth forcing function is one that is not continuous or differentiable at certain points. This can make it more challenging to solve the associated ODE, as traditional methods may not be applicable.

4. How does the Laplace transform help with ODEs with non-smooth forcing functions?

The Laplace transform can be used to convert the non-smooth forcing function into a smoother function in the complex frequency domain. This can make it easier to solve the ODE using traditional methods or specialized techniques for solving ODEs with discontinuities.

5. What are some applications of Laplace transforms in solving ODEs with non-smooth forcing functions?

Laplace transforms have a wide range of applications in engineering and physics, including solving ODEs with non-smooth forcing functions. This can be useful in analyzing systems with sudden changes or impacts, such as in mechanical systems or electrical circuits.

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