Airy equation

Example

The Airy equation is $$y'' - z\, y = 0.$$ Its coefficients are polynomials, so every point of $\C$ is an ordinary point, and all solutions are entire functions. A power series solution $y = \sum_{k \ge 0} a_k z^k$ gives $2 a_2 = 0$ and $$(k + 2)(k + 1)\, a_{k+2} = a_{k-1} \quad \text{for } k \ge 1.$$ With $a_0 = 1$ and $a_1 = 0$ one gets $y_1 = 1 + \frac{z^3}{6} + \frac{z^6}{180} + \cdots$. With $a_0 = 0$ and $a_1 = 1$ one gets $y_2 = z + \frac{z^4}{12} + \frac{z^7}{504} + \cdots$. They are linearly independent, since their Wronskian at $0$ is $y_1(0) y_2'(0) - y_1'(0) y_2(0) = 1$.