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Wikipedia's Sophie Germain; Work in elasticity; Subsequent attempts for the Prize says:

Germain had derived the correct differential equation (a special case of the Kirchhoff–Love equation),31 but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler,18 which led to incorrect boundary conditions.

18 Gray, Mary W. (2005). "Sophie Germain". In Bettye Anne Case; Anne M. Leggett (eds.). Complexities: Women in Mathematics. Princeton University Press. pp. 68–75. ISBN 0-691-11462-5.

31Ullmann, D. (2007). "Life and work of E.F.F. Chladni". European Physical Journal ST. 145 (1): 25–32. Bibcode:2007EPJST.145...25U. doi:10.1140/epjst/e2007-00145-4. S2CID 121813715.

Question: Upon which incorrect equation of Euler did Sophie Germain rely in her work that won a prix extraordinaire from the Paris Academy of Sciences?

This thorough answer to Did Sophie Germain find a flaw in Euler's equations for elastic vibrations? goes into some detail but states only that:

It could have been one of Euler's boundary conditions for the vibrating rod extended to plates. But, according to Hill, already in 1813 the Academy was impressed enough with Germain's predictions of nodal lines and frequency ratios in Chladni's experiments for square and rectangular plates, and she does not mention a boundary value issue in the 1816 memoir.

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