A symmetric polynomial in variables x_1,…,x_n is a polynomial that is invariant under arbitrary permutations of x_1,…,x_n. Polynomials are called elementary symmetric ...

An Euler equation is an ODE of the form The following is an example of an Euler equation. The Legendre linear equation is a generalization of the Euler equation. It has the ...

There are four major areas in the study of ordinary differential equations that are of interest in pure and applied science. Of these four areas, the study of exact solutions ...

Mathematica's symbolic character allows it to provide deep integrated support for algebraic numbers. At the core are Root objects, which provide exact implicit ...

Decompose[poly, x] decomposes a polynomial, if possible, into a composition of simpler polynomials.

JacobiP[n, a, b, x] gives the Jacobi polynomial P_n^(a, b)(x).

When you write down a set of simultaneous equations in Mathematica, you are specifying a collection of constraints between variables. When you use Solve, you are finding ...

The simplest type of linear second-order ODE is one with constant coefficients. This linear second-order ODE has constant coefficients. Notice that the general solution is a ...

A linear ODE with constant coefficients can be easily solved once the roots of the auxiliary equation (or characteristic equation) are known. Some examples of this type ...

A Frobenius equation is an equation of the form where a_1, …, a_n are positive integers, m is an integer, and the coordinates x_1, …, x_n of solutions are required to be ...