Physical Law and Solutions:

A physical law (e.g., general relativity) is usually expressed mathematically by a differential equation or a set of coupled (i.e., interacting) differential equations (e.g., the Einstein field equations) which are what is eternally true at every point in spacetime (within the physical law's realm of validity). The application of the differential equation(s) to particular systems are solutions of the differential equation(s). Virtually always to obtain solutions, you must supply boundary conditions (including often their special case initial conditions).
Exact analytic solutions are those you can just write down as formulae (e.g., the Schwarzschild solution to the Einstein field equations). Other kinds of solutions are approximate (by perturbation theory or otherwise) or are obtained by numerical methods on the computer.

Caption: To illustrate the solutions of differential equations, see the animations showing the projectile motion trajectories for an inclined launch with NO drag (black), Stokes law drag (blue), and Newton drag (green). Launch angle = 70° and in natural units little g = 1, launch speed v = 1.25, and terminal velocity for both drag types vmax=0.65.

  
  a = [(sum of forces)/mass]
      at every point in 
      spacetime.

The trajectories in the animations the solution is analytic and simple in the case of NO drag, analytic, but NOT so simple, in the case of Stokes law drag, and by numerical solution in the case of Newton drag which has NO exact analytic solution it seems (Wikipedia: Projectile Motion: Trajectory of a projectile with Newton drag).