I'm asking this question because of a very specific problem I ran into while tackling a PDE via a similarity solution approach, but I think it'd make more sense to ask in full generality:
Suppose we have a PDE in two variables $x$ and $y$ for one function $u$. We attempt a similarity solution of the form $u=x^\alpha f(\eta)$ where $\eta=x/y^\beta$ is our similarity variable. Now, as prompted by our problem, we plug in this form into some auxiliary PDE in order to plug the resulting expression into our main PDE, which will reduce it to an ODE (if that made no sense, don't worry about it).
Anyhow, the crux of the question is: plugging into our auxiliary PDE and changing differentiation with respect to $x$ by differentiation with $\eta$ with an appropriate change of variables, we end up with an expression of the form $$\frac{\partial u}{\partial \eta}=\frac{dG}{d\eta}(\eta)f(x)$$ Now, at this point, suppose we know that $G(\eta=0)=0$. Then, given how we've defined $\eta$, are we justified in integrating the equation from $\eta=0$ to $``\eta=\eta"$ to get $$u=G(\eta) f(x)$$ My concern is that $\eta$ is inherently related to $x$ by virtue of how we defined it. So when we integrate with respect to $\eta$, is it OK to somehow treat $x$ as a constant? Just as $x$ and $y$ were initially mutually independent variables, is it the case that we can now view $x$ and $\eta$ to be mutually independent variables?
Yes, I think this integration is OK. Try the following thought experiment. Suppose $x$ and $\eta$ are mutually independent (so far as you are concerned). Someone came along and created $y$ from $x$ and $\eta$. Now, $x$ and $y$ are not mutually independent. They then present you with the PDE of $u$ as a function of $x$ and $y$. You then unravel this back to $u$ as a function of $x$ and $\eta$.
I think the problem starts with the assumption that $x$ and $y$ are mutually independent. I find this questionable, since if they were independent, there would not have be a PDE to start with; i.e. you would not have needed a similarity transform to get to the ODE.
You found a similarity transform that reduced the PDE to an ODE. You can think of this as looking for an auxiliary variable $\eta$ such that $x$ and $\eta$ are independent with respect to $u$.