How Can Discontinuous Driving Functions Be Solved Without LaPlace Transforms?

[Skrew]

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.

 

[Dickfore]

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.

 

[Skrew]

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.