It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in optimal control ( the Hamilton-Jacobi equation ), differential games ( the Isaacs equation ) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

Assuming that the solution of this equation can be written as

The equation on the left is the Bessel equation which has the general solution

The equation on the right has the general solution

Then the solution of the acoustic wave equation is

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation.

Adding concentrated acid reverses the equation, and the copper ions go back into an aqueous solution.

Assuming that all the particles start from the origin at the initial time t = 0, the diffusion equation has the solution

Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates.

Y < sub > α </ sub >( x ) is necessary as the second linearly independent solution of the Bessel's equation when α is an integer.

* Conic bundle, an algebraic variety that appears as a solution of a Cartesian equation

Swiss physicist Felix Bloch provided a wave function solution to the Schrödinger equation with a periodic potential, called the Bloch wave.

As these methods are pushed to the limit, they approach the exact solution of the non-relativistic Schrödinger equation.

This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known.

Often, ray tracing methods are utilized to approximate the solution to the rendering equation by applying Monte Carlo methods to it.

* is a constant connected with the solution to the Lane-Emden equation.

Its importance in physics is the result of its being the solution to the differential equation describing forced resonance.

In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.

Given r, the rate of rotation is easy to infer from the constant angular momentum L, so a 2D solution can be easily reconstructed from a 1D solution of this equation.

Since the separation of variables in this case involves dividing by y, we must check if the constant function y = 0 is a solution of the original equation.

Trivially, if y = 0 then y '= 0, so y = 0 is actually a solution of the original equation.

The principal types of electrodes are: 1 ) simple solid conductors, such as discs and needles ( singles or arrays, often insulated except for the tip ), 2 ) tracings on printed circuit boards, also insulated except for the tip, and 3 ) hollow tubes filled with an electrolyte, such as glass pipettes filled with potassium chloride solution or another electrolyte solution.

In recent decades, the principal solution to the problem of the WCML's curvaceous line of route has been the adoption of tilting trains, formerly British Rail's APT, and latterly the Class 390 Pendolino trains constructed by Alstom and introduced by Virgin Trains in 2003.

Bentley's principal software solution is MicroStation.

From this principal, conservation biologists can trace communal resource based ethics throughout cultures as a solution to communal resource conflict.

A Conservative government was then formed on May 29 by Admiral Arvid Lindman, whose principal task was to find a solution of the suffrage question which both chambers could accept.

Usually, the principal manifold is defined as a solution to an optimization problem.

The principal merit of Lyapunov function-based stability analysis of ODEs is that the actual solution ( whether analytical or numerical ) of the ODE is not required.

However, the solution devised by the three men, known as the Cabinet Mission Plan, was unsatisfactory to the Indian National Congress mainly its principal leaders, and instead of having to hold together the emerging one nation, Indian National Congress leaders travelled further down the road that eventually led to Partition.

Aside from the Orthographic, six standard principal views ( Front ; Right Side ; Left Side ; Top ; Bottom ; Rear ), descriptive geometry strives to yield four basic solution views: the true length of a line ( i. e., full size, not foreshortened ), the point view ( end view ) of a line, the true shape of a plane ( i. e., full size to scale, or not foreshortened ), and the edge view of a plane ( i. e., view of a plane with the line of sight perpendicular to the line of sight associated with the line of sight for producing the true shape of a plane ).

The principal byproduct of the Solvay process is calcium chloride ( CaCl < sub > 2 </ sub >) in aqueous solution.

RenderWare's principal commercial importance was in providing an off-the-shelf solution to the difficulties of PS2 graphics programming.

As a solution to the principal – agent problem, though, tipping is not perfect.

As its principal ingredients were a mixture of laudanum ( an alcoholic solution of opium ), tincture of cannabis, and chloroform, it readily lived up to its claims of relieving pain, as a sedative, and for the treatment of diarrhea.

One of the principal uses of the notion of a subgame is in the solution concept subgame perfection, which stipulates that an equilibrium strategy profile be a Nash equilibrium in every subgame.

For moral hazard, contracting between principal and agent may be describable as a second best solution where payoffs alone are observable with information asymmmetry.

The time t and azimuthal angle φ are cyclic coordinates, so that the solution for Hamilton's principal function S can be written

The principal nth root of a positive real number A, is the positive real solution of the equation

LAMP is an acronym for a solution stack of free, open source software, referring to the first letters of Linux ( operating system ), Apache HTTP Server, MySQL ( database software, but now sometimes MariaDB ) and originally Perl ( but now sometimes PHP or Python ), principal components to build a viable general purpose web server.