Suppose I have a differential equation $$\frac{\partial\theta}{\partial t}=\frac{\partial^2\theta}{\partial x^2}$$ satisfied by $\theta(x,t)$ and I then had that $\theta(x,t)=k F(x,t)$, and I know $F(x,t)$ is dimensionless. What is a general method in this situation to answer the question "Show $F$ is a function of only the similarity variable $\eta=\eta(x,t)$"?
I am not really sure how to approach this as I don't recall ever encountering a problem like this. Any help apprecieted.
Let's take another example, the 1D wave equation: $$\frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2}=0.$$ Take $u(x,t) = U(x \pm ct)$, then $u$ satisfies the wave equation. Of course initial and boundary conditions also need to be taken into account, this will set $U$.
Can you work out an equivalent way for your (heat?) equation?