The general solution to a differential equation contains undetermined coefficients that are labeled C[1], C[2], and so on. This example has one undetermined parameter, C[1]. ...

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 ...

A partial differential equation (PDE) is a relationship between an unknown function u(x_1,x_2,…,x_n) and its derivatives with respect to the variables x_1,x_2,…,x_n. Here is ...

If the given second-order ODE is inhomogeneous, DSolve applies the method of variation of parameters to return a solution for the problem. This solves an inhomogeneous linear ...

Following are two examples of nonlinear systems of ODEs that can be solved symbolically using DSolve. The first three equations in this system of four nonlinear ODEs can be ...

When you set up mathematical formulas, you often have to introduce various kinds of local objects or "dummy variables". You can treat such dummy variables using modules and ...

Primitives which can take multiple elements. Some primitives have multi-element forms that can be processed and rendered more quickly by the Mathematica front end than the ...

Elementary transcendental functions. Mathematica gives exact results for logarithms whenever it can. Here is log_21024. You can find the numerical values of mathematical ...

"Defining Variables" discussed assignments such as x=y which set x equal to y. Here we discuss equations, which test equality. The equation x==y tests whether x is equal to ...

The built-in Mathematica iteration functions such as Table and Sum evaluate their arguments in a slightly special way. When evaluating an expression like Table[f,{i,i_max}], ...