PDEs from conservation laws
Where do PDEs come from in the first place? They are not pulled from the air — most of the famous ones are the precise mathematical statement of a single, almost obvious bookkeeping idea: stuff is conserved. Whatever quantity you track — mass, energy, charge, momentum, the number of cars on a road — the amount inside any region can only change because some of it flowed across the boundary, or because a source created or destroyed some inside. Write that balance carefully and a PDE falls out.
Here is the derivation in plain steps, the one behind half the equations in this subject. Let u be the density of some conserved quantity (amount per unit length, area, or volume), and let q be its flux (the rate it flows past a point). The total amount in a small interval changes only by what flows in at one end minus what flows out at the other, plus any source. Taking the interval small turns this balance into the conservation law u_t + q_x = 0 (or u_t + div q = f with a source f in higher dimensions). To finish, you need a constitutive relation — a physical rule giving the flux in terms of u. If the flux is just u carried along at speed c, q = c u, you get the transport equation u_t + c u_x = 0. If the flux is proportional to the negative gradient (things flow from high to low — Fourier's law for heat, Fick's law for diffusion), q = -k u_x, you get the heat equation u_t = k u_xx. If the flux depends on u itself, q = u^2/2, you get the nonlinear Burgers equation u_t + u u_x = 0.
This single template — balance plus a flux law — generates an astonishing fraction of mathematical physics, and seeing it changes how you read every equation. The other great source is variational principles: many equations arise because nature minimises something (energy, action, area), and the condition for a minimum is itself a PDE (the Euler-Lagrange equation) — for instance, a soap film minimises area and obeys the minimal surface equation, and Laplace's equation is the Euler-Lagrange equation for minimising the energy integral of grad u squared. Conservation laws and variational principles are the two great wellsprings from which PDEs flow; recognising which one is at work tells you a great deal about an equation before you have solved anything.
From conservation u_t + q_x = 0 plus a flux law: q = c u gives transport u_t + c u_x = 0; q = -k u_x (Fourier/Fick) gives heat u_t = k u_xx; q = u^2/2 gives Burgers u_t + u u_x = 0. The conservation law is the same; the flux law decides which equation you get.
One balance law, three flux relations, three classic equations — linear transport, diffusion, and a nonlinear shock-former.
The conservation law u_t + q_x = 0 only becomes a closed PDE once you add a constitutive (flux) law — and that law is a modelling assumption, not a theorem. Different flux laws give very different equations from the identical conservation principle, and a poorly chosen flux law gives a wrong model even though the bookkeeping is correct.