1. The Airy equation is a second-order linear differential equation defined as: $$ y'' - xy = 0 $$ where $y''$ is the second derivative of $y$ with respect to $x$. 2. This equation arises in physics and engineering, particularly in quantum mechanics and wave propagation near turning points. 3. Solutions to Airy's equation are called Airy functions, typically denoted as $\mathrm{Ai}(x)$ and $\mathrm{Bi}(x)$. 4. These solutions behave differently as $x$ changes, with $\mathrm{Ai}(x)$ tending to zero for large positive $x$, and $\mathrm{Bi}(x)$ growing exponentially in that region. 5. Understanding these functions is useful for analyzing scenarios where standard approximations fail and near critical points of evolution in systems.