Transfer Function: State Space for DC motor
You have a two state system, so (generically) you cannot replace the coupled
two state system (Equation (8) in your link) by a single first order differential equation.

However, since the system is completely controllable, you can transform the system into controller canonical form and write the system as a single second order differential equation. In this case we can read off the equation from 
the transfer function.

You can see from Equation (7) that we have the system
\$ ((J s+b)(Ls + R) + K^2) \hat{\omega} = K \hat{V}\$, where
\$\hat{\omega}, \hat{V}\$ are the Laplace transforms of the rotational speed and
input voltage respectively. This gives the following differential equation
\$ JL \ddot{\omega}(t) + (JK+bL)\dot{\omega}(t) + (K^2+bK) \omega(t) = K v(t) \$.

It is not clear that this is more convenient that the state space form, but perhaps gives more insight into the rotational dynamics.