Sample software implementations:

Convergence: math.stackexchange.com/questions/629630/simple-proof-euler-mascheroni-gamma-constant

The name is a bit obscure if you don't think in very generalized terms right out of the gate. It refers to a linear polynomial of multiple variables, which by definition must have the super simple form of:and then we just put the unknown $y$ and each derivative into that simple polynomial:except that now the $c_i$ are not just constants, but they can also depend on the argument $x$ (but not on $y$ or its derivatives).

Explicit solutions exist for the very specific cases of:

Order of the highest derivative that appears.

Technique to solve partial differential equations

Naturally leads to the Fourier series, see: solving partial differential equations with the Fourier series, and to other analogous expansions:

One notable application is the solution of the Schrödinger equation via the time-independent Schrödinger equation.

Bibliography:

The finite element method is one of the most common ways to solve PDEs in practice.

Used for example in FreeFem and FEniCS Project as the input description of the PDEs, TODO why.

Used to solve partial differential equation.

TODO understand, give intuition, justification of bounds and JavaScript demo.

The majority likely comes from physics: