Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over $\Bbb{C}(z)$ and $G^0$ is its connected component. It would be helpful if someone could give me some reference for this fact.
2026-03-26 11:00:17.1774522817
Airy differential equation and Galois group
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This differential Galois group $G$ of the Airy equation is $SL_2$. Therefore it is connected, i.e. $G = G^0$.
It is computed for example in Examples 4.29 of [1]. See also example 6.21.
[1] Andy R. Magid. Lectures on differential Galois theory, volume 7 of University Lecture Series. American Mathematical Society, Providence, RI, 1994.