My understanding of differential equations (DEs) suggests the following pattern between the DE and its initial/boundary conditions: Roughly, the number of derivatives equals the number of required conditions. Except in elliptic PDEs.
More specifically:
For ODEs, the number of conditions equals the largest order of derivative. Depending on if the domain is bounded or not, these can be interpreted as boundary values or initial values.
For first order PDEs, we require one condition per each variable. If a set of variables are bounded, we can have a single condition for the boundary of this domain (i.e. boundary condition). If a variable is on a semi-infinite domain (e.g. time) we have a single condition (initial value). If we have an infinite domain, then we typically require some type of asymptotic behavior.
For the parabolic equations, we require one condition due to the one “time” derivative and two conditions for the "spatial" variables.
For hyperbolic equations, we require two conditions due to the two "time" derivatives and two conditions for the two "spatial" derivatives.
But for elliptic equations, due to the use of Green's functions, we need only a single boundary condition. There are two "spatial" derivatives though.
From the answers to this question it seems the pattern should not hold for elliptic equations. But then, why?
From the answers to this question it's probably some deeper issue related to well-posedness.
But, if we have second order derivatives, don't we need to integrate twice and therefore are required to have two free constants to set?
