Fermat always started with an algebraic equation and then described the geometric curve which satisfied it, while Descartes starts with geometric curves and produces their equations as one of several properties of the curves.

This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 0 specifies only the single point ( 0, 0 ).

This equation always has one solution that is not related to a physical angle, and two that are.

where n is a given nonsquare integer and integer solutions are sought for x and y. Trivially, x = 1 and y = 0 always solve this equation.

This equation reflects the fact that the position and velocity vectors of a point traveling on the sphere are always orthogonal to each other.

Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency corresponding to the energy of the transition.

Quintessence differs from the cosmological constant explanation of dark energy in that it is a dynamic equation that changes over time, unlike the cosmological constant which always stays constant throughout time.

and if is chosen so that the constraint equation will be always satisfied.

The detailed distribution of the diffracted light depends on the detailed structure of the grating elements as well as on the number of elements in the grating, but it will always give maxima in the directions given by the grating equation.

However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true ; for instance, he shows that the equation x < sup > 4 </ sup > = 4x − 3, although incomplete, has four solutions ( counting multiplicities ): 1 ( twice ), − 1 + i √< span style =" text-decoration: overline "> 2 </ span >, and − 1 − i √< span style =" text-decoration: overline "> 2 </ span >.

In the formulation of the De Broglie – Bohm theory, there is only a wave function for the entire universe ( which always evolves by the Schrödinger equation ).

Unlike the universal wave function, the conditional wave function of a subsystem does not always evolve by the Schrödinger equation, but in many situations it does.

The fact that the conditional wave function of a subsystem does not always evolve by the Schrödinger equation is related to the fact that the usual collapse rule of Standard Quantum Theory emerges from the Bohmian formalism when one considers conditional wave functions of subsystems.

The value of this equation is always a three figure number.

The advent of quantum decoherence theory allowed alternative approaches ( such as the Everett many-worlds interpretation and consistent histories ), wherein the Schrödinger equation is always satisfied, and wavefunction collapse should be explained as a consequence of the Schrödinger equation.

However, iconographically, such an equation is questionable, since what is considered the Classic Maya moon goddess, identifiable through her crescent, is always represented as a fertile young woman.

This equation always has a unique solution, and in the parallel-field approximation, it is compatible with the torque requirement.

Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation ( s ) is solvable.

Metropolis Light Transport is an unbiased method that, in some cases ( but not always ), converges to a solution of the rendering equation quicker than other unbiased algorithms, path tracing and bidirectional path tracing.

This may be contrasted with the vector cross product, which does: in a three-dimensional vector space, the three vectors in the equation will always form a right-handed set ( or a left-handed set, depending on how the cross product is defined ), thus fixing an orientation in the vector space.

Everett claims that the universe has a single quantum state, which he called the universal wavefunction, that always evolves according to the Schrödinger equation or some relativistic equivalent ; now the measurement problem suggests the universal wavefunction will be in a superposition corresponding to many different definite macroscopic realms (" macrorealms "); that one can recover the subjective appearance of a definite macrorealm by postulating that all the various definite macrorealms are actual – it seems to each observer that " we just happen to be in one rather than the others " because " we " are in all of them, but each are mutually unobservable.

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.

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

The equation on the right has the general solution

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

Bessel himself originally proved that for non-negative integers n, the equation J < sub > n </ sub >( x ) = 0 has an infinite number of solutions in x.

Computational chemists often attempt to solve the non-relativistic Schrödinger equation, with relativistic corrections added, although some progress has been made in solving the fully relativistic Dirac equation.

Solving the hydrostatic equation then leads to a model white dwarf which is a polytrope of index 3 / 2 and therefore has radius inversely proportional to the cube root of its mass, and volume inversely proportional to its mass.

The equation of a catenary in Cartesian coordinates has the form

When these forces are added, the equation of motion has the form:

From the Shockley ideal diode equation given above, it might appear that the voltage has a positive temperature coefficient ( at a constant current ), but usually the variation of the reverse saturation current term is more significant than the variation in the thermal voltage term.

Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

The wave that emerges from a point source has amplitude at location r that is given by the solution of the frequency domain wave equation for a point source ( The Helmholtz Equation ),

The Drake equation may furthermore be multiplied by how many times an intelligent civilization may occur on planets where it has happened once.

Thus, if n < sub > r </ sub > is the average number of times a new civilization reappears on the same planet where a previous civilization once has appeared and ended, then the total number of civilizations on such a planet would be ( 1 + n < sub > r </ sub >), which is the actual reappearance factor added to the equation.

Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems.

" Stated in more modern language, " The equation a < sup > n </ sup > + b < sup > n </ sup > = c < sup > n </ sup > has no solutions for any n higher than two.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation.

** a linear equation has degree one,

** quadratic equation has degree two,