ODE made of Green's functions

Question

I need to solve the following equation $$\dot G + G =\delta (t)$$ $$G(-\infty)=0$$ What could it physically mean that $G(-\infty)=0$ in the context of wave propagation?

Answer 1

Multiply with the integrating factor $$ (e^tG(t))'=e^tδ(t)=δ(t)\implies e^tG(t)=c+u(t), $$ the boundary condition implies $c=0$.

Apart from the equation being no wave equation, one might be interested in the propagation of localized waves. This implies that their amplitude falls to zero at infinity. If you compose the wave from some basis, then you might want that the basis functions have the same property.