The differential equation 𝑄=0 in which 𝑄 is a quadratic form in 𝑦”,𝑦’,𝑦 having meromorphic coefficients

Chalkley, Roger

doi:10.1090/S0002-9939-1992-1112488-7

The differential equation $Q=0$ in which $Q$ is a quadratic form in $y”,y’,y$ having meromorphic coefficients
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Proc. Amer. Math. Soc. 116 (1992), 427-435 Request permission

Abstract:

A simple necessary and sufficient condition is given for the solutions of $Q = 0$ to be free of movable branch points. And, when the condition is satisfied, all the solutions of $Q = 0$ can be obtained by solving linear differential equations of order $\leq 2$. There are four mutually exclusive cases. We shall relate Case 4 to less convenient conditions P. Appell had introduced. We shall also show how Cases 3 and 4 together motivated our discovery of an identity that is essential for a satisfactory theory of relative invariants for homogeneous linear differential equations.

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