Perturbing initial condition and perturbing ode

Question

Is perturbing the initial condition $x=x_0$ to $x=x_0+\delta$ for a non-autonomous system

$\dot{x}=f(x,t)$

which has zero as equilibra, same as perturbing the ode itself

$\dot{x}=f(x,t)+g(x,t)$

with $\parallel g(x,t) \parallel < \delta_1$ for some $\delta_1 > 0$. In other words if we perturb the initial condition can we obtain a perturbed ode with bounded perturbation term and the otherway? If not what other condition is necessary for the same?

Answer 1

Think of

$$ \dot x + a x = 0 $$

with general solution

$$ x = C_0 e^{-a t} $$

As can be seen the initial conditions are independent from the differential equation structure. then

$$ x+\delta x = C_0 e^{-at}+\delta C_0 e^{-at} $$