Preface


This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The Mathematica commands in this tutorial are all written in bold black font, while Mathematica output is in normal font.

Finally, you can copy and paste all commands into your Mathematica notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License (GPL). You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

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Introduction to Linear Algebra with Mathematica

Airy Functions


George Airy

The following equation

\[ \frac{{\text d}^2 y}{{\text d}x^2} -x\, y(x) =0 \]
is called Airy equation, named after the British astronomer George Biddell Airy (1801--1892). This differential equation of second order has two linearly independent solutions Ai(x), the Airy function of the first kind, and Bi(x), the Airy function of the second kind. These two functions have special names in Mathematica: AiryAi[x] and AiryBi[x].

Series[AiryAi[x], {x, 0, 9}]:
N[%]
Out[2]= 0.355028 -0.258819 x + 0.0591713 x^3 - 0.0215683 x^4 +
0.00197238 x^6 - 0.000513531 x^7 + 0.0000273941 x^9 - 5.7059 x 10^^{-6} + )(x)^10

A Sturm-Liouville equation is a second-order linear differential equation of the form

\[ L[y] + \lambda\,\rho (x)\,y =0 \qquad\mbox{where}\quad L[y]= \frac{{\text d}}{{\text d}x} \left( p(x)\,\frac{{\text d}y}{{\text d}x} \right) + q(x)\,y ,\qquad x\in (a, b). \]
Usually, the coefficients \( p(x) >0, \quad \rho (x) >0 , \) are positive and the function \( q(x) \) is real-valued. If endpoints x = a, b are finite, then some boundary conditions are imposed at these points so that the differential operator L becomes self-adjoint.

Airy functions are solutions to the Airy differential equation
y''[x] - x y[x] == 0
There are two linearly independent solutions, called by Mathematica as
AiryAi[x] and AiryBi[x].

Series[AiryAi[x], {x, 0, 9}]:
N[%]
Out[2]= 0.355028 -0.258819 x + 0.0591713 x^3 - 0.0215683 x^4 +
0.00197238 x^6 - 0.000513531 x^7 + 0.0000273941 x^9 - 5.7059 x
10^^{-6} + )(x)^10

Example:

 

 

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