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The Fredholm integral equation of the second kind f(x)=1+1/piint_(-1)^1(f(t))/((x-t)^2+1)dt that arises in electrostatics (Love 1949, Fox and Goodwin 1953, and Abbott 2002).

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 ...

y^('')-mu(1-1/3y^('2))y^'+y=0, where mu>0. Differentiating and setting y=y^' gives the van der Pol equation. The equation y^('')-mu(1-y^('2))y^'+y=0 with the 1/3 replaced by ...

Given a system of two ordinary differential equations x^. = f(x,y) (1) y^. = g(x,y), (2) let x_0 and y_0 denote fixed points with x^.=y^.=0, so f(x_0,y_0) = 0 (3) g(x_0,y_0) ...

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 expression is called "well-defined" (or "unambiguous") if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to not be ...

Spinor fields describing particles of zero rest mass satisfy the so-called zero rest mass equations. Examples of zero rest mass particles include the neutrino (a fermion) and ...

The most general forced form of the Duffing equation is x^..+deltax^.+(betax^3+/-omega_0^2x)=gammacos(omegat+phi). (1) Depending on the parameters chosen, the equation can ...

(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+alpha^2y=0 (1) for |x|<1. The Chebyshev differential equation has regular singular points at -1, 1, and infty. It can be solved by series ...

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 ...