# Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods

Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods

Consider the Airy differential equation, , where , , and . Values of and (the eigenvalues and eigenfunctions) can be determined by solving the generalized eigenvalue problem , where the matrices and are given in the details section. The eigenfunction is given by , where is the classic Airy function and is the eigenvalue. This Demonstration approximates the values of the eigenvalues and eigenfunctions (up to ) numerically using spectral methods. When the number of grid points is large, the numerical values of at the grid points match the Airy function very closely.

u''=λxu

u≠0

u(±1)=0

-1≤x≤1

λ

u

Au=λBu

A

B

n

th

Aiλx

n

1/3

Ai(x)

λ

n

n

th

n=10

u