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

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The discussion focuses on solving the ordinary differential equation (ODE) y''(t) = x''(t) with x(t) as the ramp function, emphasizing the need to address the discontinuity in x'(0) when applying the Laplace transform. Participants highlight the importance of understanding how the Laplace transform is defined, particularly in relation to singularities and discontinuities at the origin. It is noted that the right-hand side of the problem can be interpreted as a delta function, which complicates the transformation process. The conversation suggests that different definitions of the Laplace transform may lead to varying approaches in handling such discontinuities. Ultimately, clarity on the treatment of these mathematical nuances is crucial for accurate problem-solving.
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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
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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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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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