Question about Controlled Integrator examples


The examples I have question about is here:
Controlled Integrator example

The controlled integrator examples construct the system based upon the following equation:

$\dot{a}(t)=control(t) \cdot a(t)+B \cdot input(t)$

The value of B is not given directly in the examples, but the first example does say the equivalent NEF equation for this integrator is:

$\dot{a}(t)=control(t) \cdot a(t)+\tau \cdot input(t)$

I know this is a Time-varying system, but I’m still not sure how the equivalent NEF equation is derived. Can someone help me?