Assuming that $dS(t)=rS(t) \, dt+\sigma S(t) \, dW(t)$, and $dW(t)\sim N(0,\Delta t)$, and having a portfolio where $\Pi = C - \alpha S$ then the Black-scholes PDE is:
$$\frac{\partial C}{\partial t} +rS\frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2\frac{\partial^2 C}{\partial S^2}-rC=0,$$
Which results from setting:
$$\left(\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}\right) \, \Delta t$$ equal to the bank rate of $\Delta\Pi = r\Pi\,\Delta t.$
My question is, is it ever the case that we do not set the second equation equal to $\Delta\Pi = r\Pi\,\Delta t$? How would that affect the Black-scholes pde? For example, what if $\Pi =C - P - \gamma S$?