Euler, Heun, Midpoint, and RK4 are all straightforward explicit solvers.
RK45 is adaptive explicit solver.
Adams Bashforth is an iterative solver. It's broken at the moment. Someone give me a hint.
Backwards Euler and Trapezoid Implicit are implicit solvers for which the math was solved per-problem. (Oh yes, I have a Computer Algebra System in the works too). Backwards Euler and Trapezoid Newton are based on a Newton root-finding if no explicit calculations are provided. This rootfinding assumes the change in next function wrt the next x is zero. It produces a simple fixed-point convergence for derivative.
Would like to add quadrature integrators (Gaussian, Hermite, etc) and maybe, to get eccentric, add different fixed-point solvers: root finding based on several-derivative constraints rather than just linear, as Newton uses.