Consider the heat equation $$k\nabla^2u=k\left ( \frac{\partial^2u }{\partial x^2}+ \frac{\partial^2u }{\partial y^2} \right )=\frac{\partial u}{\partial t}$$ in two dimensions, where $k$ is a constant. By making the substitution $s=kt$, I want to show that this equation can be written as $$\frac{\partial^2u }{\partial x^2}+ \frac{\partial^2u }{\partial y^2} =\frac{\partial u}{\partial s}.$$ My initial thought was to just substitute directly using the fact that $$k=\frac{s}{t}.$$ However, this did not get me far. I know I have to use the chain rule somehow, but right now I am not really seeing it. Any thoughts?
Thanks in advance!
Use $$\frac{\partial u}{\partial s}=\frac{\partial u}{\partial t}\frac{\partial t}{\partial s}$$ Here, $\frac{\partial t}{\partial s}=1/k$, so $$\frac{\partial u}{\partial s} = \frac{1}{k}\frac{\partial u}{\partial t}$$ Substitute for $\frac{\partial u}{\partial t}$, and the answer follows.