The general form of a nonlinear second-order ODE is For simplicity, assume that the equation can be solved for the highest-order derivative y^ ′′(x) to give There are a few ...

DSolve returns a general solution for a problem if no initial or boundary conditions are specified. The general solution to this equation is returned. However, if initial or ...

Eigenvalues and eigenvectors. The eigenvalues of a matrix m are the values λ_i for which one can find nonzero vectors v_i such that m.v_i λ_iv_i. The eigenvectors are the ...

When you ask for the square root s of a number a, you are effectively asking for the solution to the equation s^2a. This equation, however, in general has two different ...

If you have an equation like 2x==0, it is perfectly clear that the only possible solution is x->0. However, if you have an equation like ax==0, things are not so clear. If a ...

Mathematica provides built-in support for both programmatic and interactive image processing, fully integrated with Mathematica's powerful mathematical and algorithmic ...

If you type in an expression like 0/0, Mathematica prints a message, and returns the result Indeterminate. An expression like 0/0 is an example of an indeterminate numerical ...

Whenever machine-precision numbers appear in a calculation, the whole calculation is typically done in machine precision. Mathematica will then give machine-precision numbers ...

Matrix inversion. Here is a simple 2×2 matrix. This gives the inverse of m. In producing this formula, Mathematica implicitly assumes that the determinant ad-bc is nonzero.

A differential system can sometimes be solved by analytic means. The function DSolve implements many of the known algorithmic techniques. However, differential systems that ...