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 ...

Mathematical constants. Euler's constant EulerGamma is given by the limit γUnderscriptBox[lim, RowBox[{m, , ∞}], LimitsPositioning -> True](UnderoverscriptBox[∑, RowBox[{k, ...

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

UseFrontEnd[expr] evaluates expr in an environment where the kernel can make use of the services of the notebook front end.

MLINK getMathLink (WolframLibraryData libData) gets a MathLink connection to use for evaluations in Mathematica.