The symbol := means that the thing on the colon side is being defined as what's on the equals side. So the notation there is merely that
$\displaystyle \frac{du}{dt}$ is entirely equivalent to $\displaystyle d_{t}u,$ which is entirely equivalent to $\displaystyle \dot{u}.$
The actual differential equation is
$\displaystyle \dot{u}=g(t)u+h(t),\;u(0)=u_{0}.$
That, and the hint provided should get you started. What ideas do you have now?