Answer
$3.71728$
Work Step by Step
In order to determine the exact solution, isolate the x and y terms on one side and integrate both sides.
$\int dy=\int 2xe^{x^2} dx$ ...(1)
In order to determine the differential equation, we will have to take the derivative of the differential equation.
Equation (1) gives: $y=e^{x^2}+c$ ...(2)
Apply the initial conditions, to calculate the value of $c$
$2=e^{0^2}+c \implies c=1$
Now, equation (2) becomes:
$y=e^{x^2}+1$
Thus, the exact solution is: $y(1)=e^{(1)^2}+1 \approx 3.71728$
