I have found the following theorem online: If the functions p and g are continuous on the interval $I: \alpha < t < \beta$ containing the point $t = t_0$, then there exists a unique function $y = \gamma(t)$ that satisfies the differential equation
$$y'(t)+p(t)y=g(t)$$
for each $t \in I$, and that also satisfies the initial condition $$ y(t_0)=y_0$$ where $y_0$ is an arbitrary prescribed initial value.
But I need to have a valid reference for it. So do anyone know some book, where it is stated and proved, such that I can reference it.