Airy’s equation

Definition

Airy's equation is a second-order linear differential equation of the form $y'' - xy = 0$. It is notable for its solutions known as Airy functions, which are particularly useful in physics.

5 Must Know Facts For Your Next Test

1. Airy's equation is written as $y'' - xy = 0$.
2. The general solution to Airy's equation involves two linearly independent solutions, typically denoted as $Ai(x)$ and $Bi(x)$.
3. Airy functions can be expressed using power series expansions.
4. In the context of Taylor Series, the coefficients of the series for $Ai(x)$ and $Bi(x)$ can be determined by solving recurrence relations derived from substituting the series into Airy's equation.
5. Airy's equation appears in quantum mechanics, optics, and other fields involving wave propagation.

Review Questions

• What is the general form of Airy's equation?
• How are the solutions to Airy's equation typically denoted?
• Describe how power series can be used to find solutions to Airy's equation.

Related terms

Taylor Series:

A representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

Differential Equation:

An equation involving derivatives of a function or functions.

$Ai(x)$ and $Bi(x)$: $Ai(x)$ and $Bi(x)$ are the standard notations for the two linearly independent solutions to Airy's differential equation.