Nondimensionalizing an DE

Question

I am struggling to understand the validity of what is done when you have a differential equation with dimensional variables and you are able to turn it into a differential equation with less dimensional variables. To be specific, suppose you do it for the differential equation on the picture attached. I am not sure what is being done here, I don’t know if the notation of these notes is a little ambiguous but I don’t see how they are using the chain rule. Please, can someone explain it to me with unambiguous notation and not weird nonsensical differentials?

Answer 1

Aside from the fact that multiplicative constants factor from derivatives, this is indeed just the chain rule: $$\frac{dc}{dt}=\frac{dc}{d\tilde{t}}\frac{d\tilde{t}}{dt}=\frac{d(C_c \tilde{c})}{d\tilde{t}} \frac{d(t/T_c)}{dt} = \frac{C_c}{T_c}\frac{d\tilde{c}}{d\tilde{t}}.$$ You could also write this as $\dfrac{dc}{dt}=\dfrac{dc/d\tilde{t}}{dt/d\tilde{t}}=\dfrac{C_c}{T_c}\dfrac{d{\tilde c}}{d\tilde{t}}$.