I wanted to know if Omar Khayyam did work on ODE and if there is any connection between that and the cubic equations.
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While elements of Newton's Principia draws upon the classical method of exhaustion from the Greeks, as well as some contemporary work of other mathematicians (e.g. John Wallis' Arithmetica Infinitorum), there is no known record of calculus having been discovered/invented (depending on your point of view) independently prior to the, say, 13th century.
Based on that, the known facts do not point toward Omar Khayyam having even the language in which to write down a differential equation, let along doing work in it.