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.
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I could not figure out the math equation. The new data did not fit the existing equation. An equation can be a math formula or standard method.
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figure math means figure math
You have to figure that out! Variables, in math, are there because you need to find them and figure the problem out. Usually, on the side it tells you what the variables mean, or you have to find that variable by making the equation make sense if they give u the answer.
I think you are referring to checking a math equation. After you solve an equation you should go back and check your work to make sure you got the right answer. You can do this by plugging your answer back into the equation