How Many BC and IC Do We Need for Different Equations?

[Glass]

So I'm confused on how we know we need x many BC and/or IC. For example, regarding the heat equation, we need 2 BC, and 1 IC (supposedly because there are two spatial derivatives and one time derivative). And similarly, for the Wave equation we need 2 BC and 2 IC. And another thing, why is it that when we look at D'Alembert's solution of the infinite string wave equation, why we only need 2 IC and no IC? Thanks, I'm very confused.

 

[Ben Niehoff]

Each integration supplies an arbitrary constant (or in partial differential equations, an arbitrary function of the other independent variables). Thus, for the most general solution, there are N extra degrees of freedom for each Nth-order derivative (in the heat equation, 2 for space + 1 for time; in the wave equation, 2 + 2, etc.). Therefore, in order to get to an equation that describes a specific physical situation, we need to supply additional conditions to narrow down these extra degrees of freedom.

As for exactly which additional conditions make the most sense, that depends on the physical situation. Often, we are interested in the time evolution of a system, and so it makes sense to give an initial condition for every order in the time derivatives. But it is not strictly necessary that it be so; we could, for example, give final conditions instead, if we were interested in calculating things that way.