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Which of the following is a solution to the differential equation $\cfrac{dy}{dx}=2xy$ with an initial condition of $\left ( -1,e^{2} \right )$?

$y=e^{x^{2}}$

$y=1+e^{x^{2}}$

$y=e^{x^{2}+1}$

$y=-e^{x^{2}+1}$