Further Examples of Exactly Solvable ... (Mathematica Tutorial)

The solutions to many second-order ODEs can be expressed in terms of special functions. Solutions to certain higher-order ODEs can also be expressed using AiryAi, BesselJ, ...

Linear Second-Order ODEs with Rational ... (Mathematica Tutorial)

The hypergeometric functions play a unifying role in mathematical analysis since many important functions, such as the Bessel functions and Legendre functions, are special ...

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 ...

Evaluation in Patterns, Rules, and ... (Mathematica Tutorial)

There are a number of important interactions in Mathematica between evaluation and pattern matching. The first observation is that pattern matching is usually done on ...

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 ...