Nonlinear Second-Order ODEs (Mathematica Tutorial)

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 ...

References (Mathematica Tutorial)

[AB04] Abell, M. L. and J. P. Braselton. Differential Equations with Mathematica. (3rd ed.) Elsevier Academic Press, 2004. [A89] Abramov, S. A. "Rational Solutions of Linear ...

Second-Order PDEs (Mathematica Tutorial)

The general form of a linear second-order PDE is Here uu(x,y), and a, b, c, d, e, f, and g are functions of x and y only—they do not depend on u. If g0, the equation is ...

Is the Problem Well-Posed? (Mathematica Tutorial)

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 (Mathematica Tutorial)

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 ...

Entering Two-Dimensional Input (Mathematica Tutorial)

When Mathematica reads the text x^y, it interprets it as x raised to the power y. In a notebook, you can also give the two-dimensional input x^y directly. Mathematica again ...

Functions That Do Not Have Unique Values (Mathematica Tutorial)

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 ...

Generic and Non-Generic Solutions (Mathematica Tutorial)

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 ...

Indeterminate and Infinite Results (Mathematica Tutorial)

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 ...

Labeling Three-Dimensional Graphics (Mathematica Tutorial)

Mathematica provides various options for labeling three-dimensional graphics. Some of these options are directly analogous to those for two-dimensional graphics, discussed in ...