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 ...

There are some close connections between finding a "local minimum" and solving a set of nonlinear equations. Given a set of n equations in n unknowns, seeking a solution r(x) ...

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 ...

The solutions to many second-order ODEs can be expressed in terms of special functions. Solutions to certain higher-order ODEs can also be expressed using AiryAi, BesselJ, ...

Dt[f, x] gives the total derivative d f/d x. Dt[f] gives the total differential d f. Dt[f, {x, n}] gives the multiple derivative d^n f/d x^n. Dt[f, x_1, x_2, ...] gives d/d ...

The Lotka–Volterra system arises in mathematical biology and models the growth of animal species. Consider two species where Y_1(T) denotes the number of predators and Y_2(T) ...

Total differentiation operations. When you find the derivative of some expression f with respect to x, you are effectively finding out how fast f changes as you vary x. Often ...

Unicode: F74C. Alias: Esc dd Esc. Compound operator with built-in evaluation rules.  can only be interpreted by default when it appears with ∫ or other integral operators. ∫ ...

FindFit[data, expr, pars, vars] finds numerical values of the parameters pars that make expr give a best fit to data as a function of vars. The data can have the form {{x_1, ...

Mathematica 7 represents another major achievement in Mathematica's long history of innovation in mathematics and algorithms. Building on the broad capabilities of ...