My question is about the content in these notes of the MIT OCW course on differential equations, 18.03. (anyways, I will try to keep it self contained) It is about the domain of the solution when applying the Laplace transform in ODEs with rest initial conditions.
First of all, let $\lim_{t\to 0^-} t = 0^-$.
It is stated that the "rest initial conditions" of some differential equation $p(D)x=f(t)$ means that $x(0^-) = 0$, $\dot{x}(0^-) = 0$, ... and $x(t)=0$ for any $t<0$.
So consider, for example, the DE $$\dot{x}+3x = e^{-t}$$ After solving the differential equation with the Laplace transform (I know there are easier methods, but these notes cover the application of the Laplace transform to solve DEs), we arrive to $$ x(t) = \frac{1}{2}e^{-t}-\frac{1}{2}e^{-3t}, t>0$$ The things I don't understand are:
$(1)$ How the initial conditions can be true, since for $t<0$, $x(t) = 0$ and so the left side of the equation is $0$ for all $t<0$, but the right side is never zero for $t<0$, so how can this be true?
$(2)$ Can this solution be extended to the rest of the real line or are solutions via Laplace transform only defined for $t>0$?