There is a strong connection between the implicit function theorem and the theory of differential equations. This is true even from the historical point of view, for Picard's iterative proof of the existence theorem for ordinary differential equations inspired Goursat to give an iterative proof of the implicit function theorem (see Goursat [Go 03]). In the mid-twentieth century, John Nash pioneered the use of a sophisticated form of the implicit function theorem in the study of partial differential equations.
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