Consider the linear equation

with initial conditions \(x(t_{0}) = x_{0}\) and \(x'(t_{0}) = v_{0}\text{.}\) If \(g(t)\) is continuous on some interval \((t_{1}, t_{2})\) containing \(t_{0}\text{,}\) then there exists a unique solution \(x(t)\) that is continuous on \((t_{1}, t_{2})\text{.}\)