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For a second-order ordinary differential equation, y^('')+p(x)y^'+q(x)y=g(x). (1) Assume that linearly independent solutions y_1(x) and y_2(x) are known to the homogeneous ...

The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted T_n(x). They are ...

A determinant used to determine in which coordinate systems the Helmholtz differential equation is separable (Morse and Feshbach 1953). A determinant S=|Phi_(mn)|=|Phi_(11) ...

A boundary value problem is a problem, typically an ordinary differential equation or a partial differential equation, which has values assigned on the physical boundary of ...

Ellipsoidal harmonics of the second kind, also known as Lamé functions of the second kind, are variously defined as F_m^p(x)=(2m+1)E_m^p(x) ...

The surface area of a spherical segment. Call the radius of the sphere R, the upper and lower radii b and a, respectively, and the height of the spherical segment h. The zone ...

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis. Unfortunately, there are a number of ...

An orthogonal coordinate system is a system of curvilinear coordinates in which each family of surfaces intersects the others at right angles. Orthogonal coordinates ...

The four-dimensional version of the gradient, encountered frequently in general relativity and special relativity, is del _mu=[1/cpartial/(partialt); partial/(partialx); ...

The study, first developed by Boole, of shift-invariant operators which are polynomials in the differential operator D^~. Heaviside calculus can be used to solve any ordinary ...