This is part of a large exercise about dimensional analysis. Basically we have the 1-D heat equation in a rod with infinite length: $$\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2}.$$ As an initial profile we take a distribution which is everywhere vanishing except for a peak in the origin: $$u(x,0) = u_0\delta(x),$$ where $\delta(x)$ is the delta function.
I have already used dimensional analysis and found two dimensionless quantities: $$q_1 = \frac{x^2}{\kappa t}, \hspace{0.5cm} q_2 = u \frac{\sqrt{\kappa t}}{u_0}.$$ In view of the expected behavior of the system we may write $$q_2 = f^*(q_1)$$ for some function $f^*(q_1)$. Using this I have rewritten the heat equation in terms of $q_1,q_2$ and $f^*(q_1)$: $$4q_1\frac{\partial^2 f^*}{\partial q_1^2} + (q_1+2)\frac{\partial f^*}{\partial q_1} + \frac{1}{2}f^* = 0.$$ I have checked that $$f^* = c e^{-\frac{1}{4}q_1}$$ is a solution for any constant $c$. Now comes the part I don't get. It says:
Which conservation principle can be invoked to determine $c$? Check that the dimensional solution reads as $$u(x,t) = \frac{u_0}{\sqrt{4\pi\kappa t}}e^{\frac{-x^2}{4\kappa t}}.$$
I figure that the conservation principle must be the conservation of energy (heat) in the rod. What I don't understand is how to apply this to the dimensionless equation. Help would be much appreciated.