The Airy differential equation is often one of the first nontrivial differential equations used to demonstrate the infinite series method for determining a solution. In my own classes I use it to illustrate series techniques near an ordinary point. However, I usually do not emphasize that special functions are often defined as the solutions to specific classes of differential equations. In the brief note attached below, the Airy functions Ai(x) and Bi(x) are explored in more detail. The Bessel differential equation is also an important source of examples in ODE courses, usually to illustrate the procedure for finding Frobenius-type solutions. In fact, the Airy functions can be expressed in terms of the solutions to the Bessel differential equation of order 1/3 and the modified Bessel differential equation of order 1/3. This relationship is also verified in the note.
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