I have the following equation that I wish to represent in the form of a control system block diagram:
$a \frac{d x(t)}{dt} + b(t)x(t) = u(t)$
The variable $b(t)$ and $u(t)$ are both inputs that will change with time.
If I take the Laplace transform I get: $B(s) + a s X(s) = U(s)$
I am interested in the transfer function that describes $\frac{X(s)}{U(s)}$ but I cannot separate variables to get this...
How can I describe the model above in either a transfer function or state space model? I'd like to simulate it as it is part of a control system I'm interested in.
I assume that you define $B(s)$ as $\mathcal{L}\{b(t)\}(s)$. But that term does not occur by itself. It only occurs as a product with $x(t)$. Even if you would try to calculate the Laplace transform of the product of $x(t)$ and $b(t)$, then you still get $\mathcal{L}\{b(t)\,x(t)\}(s)\neq X(s)\,B(s)$.
This is because multiplication in the time domain results in convolution in the frequency domain (and vice versa).
Your system is not time invariant, because $b(t)$ is a function of time. You can only really define transfer function for linear time invariant (LTI) systems.
If $b(t)$ is periodic ($b(t)=b(t+T)$) then you could discretize your system with a time step of $T$. And for that model you could formulate a discrete transfer function in the z-domain. But in the current form you can't say much in general about its behavior in the frequency domain.