In many kinds of numerical computations, it is convenient to introduce approximate functions. Approximate functions can be thought of as generalizations of ordinary ...

You can define the derivative in Mathematica of a function f of one argument simply by an assignment like f'[x_]=fp[x]. This defines the derivative of f(x) to be fp(x). In ...

Here is an example of an exact ODE. This verifies the solution. Here is a contour plot of the solution.

Here is a homogeneous equation in which the total degree of both the numerator and the denominator of the right-hand side is 2. The two parts of the solution list give ...

To begin, consider an initial value problem for a linear first-order ODE. This is a linear first-order ODE. Notice that the general solution is a linear function of the ...

When you execute a command like NotebookWrite[obj,data] the actual operation of inserting data into your notebook is performed in the front end. Normally, however, the kernel ...

An image is equivalent to its symbolic expression. You can operate on an image as you would on a symbolic expression. This input produces some disks. Type // InputForm after ...

You can use lists as tables of values. You can generate the tables, for example, by evaluating an expression for a sequence of different parameter values. This gives a table ...

The simplest way to mix text and formulas in a Mathematica notebook is to put each kind of material in a separate cell. Sometimes, however, you may want to embed a formula ...

"Gauss–Newton" and "conjugate gradient" methods use derivatives. When Mathematica cannot compute symbolic derivatives, finite differences will be used. Computing derivatives ...