Abstract
The operators corresponding to third-order equations considered in this chapter generate ideals of differential dimension (1, 3). Therefore, by Kolchin’s Theorem 2.1, these equations have a differential fundamental system containing three undetermined functions of a single argument. The remarks on the structure of the solutions of linear pde’s on page 91 apply here as well. Similar as for second-order equations, various cases differing by leading derivatives are distinguished. As opposed to second-order equations, third-order equations have virtually never been treated in the literature before.
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Schwarz, F. (2012). Solving Homogeneous Third-Order Equations. In: Loewy Decomposition of Linear Differential Equations. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1286-1_7
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