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She did not figure out a particular equation but found the set of conditions under which solutions to a class of partial differential equations would exist. This is now known as the Cauchy-Kovalevskaya Theorem.
A surface is a 2-dimensional manifold. Most surfaces are curved, in which for a small "differential" section of a surface, a plane figure is uniquely determined.
ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.
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Finite Differential Methods (FDM) are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.