A linear second-order ordinary differential equation is said to be exact if An exact linear second-order ODE is solved by reduction to a linear first-order ODE.

The following is a linear first-order ODE because both y[x] and y^ ′[x] occur in it with power 1 and y^′[x] is the highest derivative. Note that the solution contains the ...

Since many functions in Mathematica give solutions in the form of rules, you need to be able to use these rules to explore and interpret your results. Although many of the ...

PadeApproximant[expr, {x, x_0, {m, n}}] gives the Padé approximant to expr about the point x = x_0, with numerator order m and denominator order n.PadeApproximant[expr, {x, ...

Monte Carlo methods use randomly generated numbers or events to simulate random processes and estimate complicated results. For example, they are used to model financial ...

Sort[list] sorts the elements of list into canonical order. Sort[list, p] sorts using the ordering function p.

GridLines is an option for two-dimensional graphics functions that specifies grid lines.

Hue[h] is a graphics directive which specifies that objects which follow are to be displayed, if possible, in a color corresponding to hue h. Hue[h, s, b] specifies colors in ...

NumericQ[expr] gives True if expr is a numeric quantity, and False otherwise.

ListVectorPlot[array] generates a vector plot from an array of vector field values.ListVectorPlot[{{{x_1, y_1}, {vx_1, vy_1}}, ...}] generates a vector plot from vector field ...