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The second-order ordinary differential equation satisfied by the Neumann polynomials O_n(x).

The equation f(x_n|x_s)=int_(-infty)^inftyf(x_n|x_r)f(x_r|x_s)dx_r which gives the transitional densities of a Markov sequence. Here, n>r>s are any integers (Papoulis 1984, ...

A linear recurrence equation is a recurrence equation on a sequence of numbers {x_n} expressing x_n as a first-degree polynomial in x_k with k<n. For example ...

For a measurable function mu, the Beltrami differential equation is given by f_(z^_)=muf_z, where f_z is a partial derivative and z^_ denotes the complex conjugate of z.

The ordinary differential equation (y^')^m=f(x,y) (Hille 1969, p. 675; Zwillinger 1997, p. 120).

The partial differential equation u_(tt)-u_(xx)-u+u^3=0.

An equation or formula involving transcendental functions.

The system of partial differential equations S_t=SxS_(xx).

An equation of the form P(x)=0, where P(x) is a polynomial.

A second-order partial differential equation arising in physics, del ^2psi=-4pirho. If rho=0, it reduces to Laplace's equation. It is also related to the Helmholtz ...