I have included image of a problem from Oxford. I was able to do the question. However, i am stuck at the last part which asks about the physical interpretation. I don't have a good background in physics. Can you please give a hint?
It's about this integral $$\int_{0}^{L}k\frac{\partial T}{\partial y}(x,y^{*})dx=Lq^{*}.$$ I think that the answer must lie in how the the integral doesn't depend on $y^{*}$. I have this vague notion that perhaps the integral is tantamount to adding up the infinitesimal changes in T (in the y direction) at each x. Since the integral is constant it means that for any y level the total rate of change along y is a constant (although it may vary at within a y at x's, the total result is the same regardless of y).
You're right in saying that this equation represents a conservation law. If you take the derivative out of the integrand, you end up with the integration of $kT,$ i.e., the total heat of the region. It means that the rate of change of the total heat is constant along $y$.