Good morning. I have a general conceptual question regarding initial conditions and constants of integration on a piecewise function. Does the initial contidion carry throughout, or does it change from piece to piece? Image attached.

Question

Good morning. I have a general conceptual question regarding initial conditions and  constants of integration on a piecewise function. Does the initial contidion carry throughout, or does it change from piece to piece? Image attached.

anonymous · Answer

attachment

anonymous · Answer

I think the image is the way to start. I'm not sure what the initial condition should be from step to step.

anonymous · Answer

I think for a case like this (in which you're given only one condition), the initial value carries through. Otherwise you would ideally be given what are called boundary conditions (i.e. $x(a),x(b)$, and $x(c)$).

anonymous · Answer

Gotcha. So why is (or would) this be preferable to using a definite integral with the bounds set for each interval?

anonymous · Answer

Actually, I think I just answered my own question on that last one. I just ran both in Mathematica and got the same answer. It seems a lot harder to solve for C than to just use a definite integral, though.

anonymous · Answer

It could be that you're examining the rate at which current flows through a wire, for example. Suppose you have a contraption set up to provide a burst/impulse of current. The ODE that would describe this sort of situation is something like
$$\frac{dI}{dt}+\frac{R}{L}I=\frac{V}{L}$$
or something to that end (I'm no physicist).

In a more general setting, the nonhomogeneous portion of the ODE could take on any workable function, say $f(t)$, which can be defined piecewise to account for the impulse.

anonymous · Answer

NIce. I'm no physicist either (yet). :) I'm doing electromagnetism this semester, though, so I expect to see a bunch of this. Thanks for your help.