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A special case of the quadratic Diophantine equation having the form x^2-Dy^2=1, (1) where D>0 is a nonsquare natural number (Dickson 2005). The equation x^2-Dy^2=+/-4 (2) ...

The second-order ordinary differential equation (1-x^2)y^('')-2(mu+1)xy^'+(nu-mu)(nu+mu+1)y=0 (1) sometimes called the hyperspherical differential equation (Iyanaga and ...

An indicial equation, also called a characteristic equation, is a recurrence equation obtained during application of the Frobenius method of solving a second-order ordinary ...

An equation of the form f(x)=b (mod m), (1) where the values of 0<=x<m for which the equation holds are sought. Such an equation may have none, one, or many solutions. There ...

There are a number of equations known as the Riccati differential equation. The most common is z^2w^('')+[z^2-n(n+1)]w=0 (1) (Abramowitz and Stegun 1972, p. 445; Zwillinger ...

The quantum electrodynamical law which applies to spin-1/2 particles and is the relativistic generalization of the Schrödinger equation. In 3+1 dimensions (three space ...

The system of partial differential equations u_t+u_x = v^2-u^2 (1) v_t-v_x = u^2-v^2. (2)

The wave equation is the important partial differential equation del ^2psi=1/(v^2)(partial^2psi)/(partialt^2) (1) that describes propagation of waves with speed v. The form ...

An equation representing a locus L in the n-dimensional Euclidean space. It has the form L:f(x_1,...,x_n)=0, (1) where the left-hand side is some expression of the Cartesian ...

The Schrödinger equation describes the motion of particles in nonrelativistic quantum mechanics, and was first written down by Erwin Schrödinger. The time-dependent ...