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

[StretchySurface]

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?

 

[STEMucator]

Check the screenshot I attached:

 

[jasonRF]

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