It is about singular pertubation theory.
I found the following article: https://pdfs.semanticscholar.org/9399/c56a5f407d3b1037db131d17f00144ee49ba.pdf
When looking at eq. $(7)$ shouldn't there be also a part $\frac{d \varphi}{d \tau}$ after $g(\cdot)$? Why do they only replace $\epsilon \frac{d y}{d t} = \frac{d y}{d \tau}$?
As reqested, here more context (in case that the link breaks down):
The following dynamic system is considered $$\dot{x}=f(x,z,u,t,\epsilon)$$ $$\epsilon \dot{z}=g(x,z,u,t,\epsilon)$$
In singular pertubation model, the model order is reduced by neglecting the fast dynamics, hence, $\epsilon = 0$ leading to $0=g(\bar{x},\bar{z},\bar{u},t,0)$. The roots of this equation are then calculated as follows: $$\bar{z}=\varphi(\bar{x},\bar{u},t)$$ The system from above is then converted into the reduced model $$\dot{\bar{x}}=f(\bar{x},\varphi(\bar{x},\bar{u},t),\bar{u},t,0)$$
Now we transform the system in the following way: $$y=z-\varphi(\bar{x},\bar{u},t)$$
where $\epsilon \frac{d y}{d t} = \frac{d y}{d \tau}$ is used to get the following result: $$\frac{dy}{d \tau}=g(t,x,y+\varphi(t,x),0)$$ The last equation is eq. (7).