A symbolical solution of the equation corresponding to the vibro-acoustic movement described is definitely impossible, even if greatly simplifying hypotheses are adopted.

That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation.

In a mechanical situation - there are normally constraints of motion (i.e. some external structure or system preventing motion in certain directions), so using a full set of Cartesian coordinates is often unnecessary: they will be related to each other by equations corresponding to the constraints.

One common choice is a linear finite state machine, where the compressed stimuli are computed by solving linear equations corresponding to internal scan cells with specified positions in partially specified test patterns.

A Fokker-Planck equation is a deterministic equation for the time dependent probability density P(A,t) of stochastic variables A. The Fokker-Planck equation corresponding to the generic Langevin equation above may be derived with standard techniques (see for instance ref.)

The system of equations corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically.

In 1931, Andrei Kolmogorov introduced the differential equations corresponding to the time-evolution of stochastic processes that proceed by jumps, today known as Kolmogorov equations (Markov jump process) (a simplified version is known as master equation in the natural sciences).

In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions.

This field equation can be derived as the Euler-Lagrange equation corresponding to the Rarita-Schwinger Lagrangian: