Final answer:
The student is asked to solve a system of differential equations with given initial conditions, but the references provided are not relevant to the problem statement. The solution would involve deriving one of the equations, substituting and integrating to find y, then using that to solve for x, but the solution process cannot be completed with the current information.
Step-by-step explanation:
The student is tasked with finding the solutions for a system of equations with derivatives, which implies this is a problem related to differential equations. Considering the initial conditions provided (x(0)=2, y(0)=2), one should first solve for y using the given equations and the initial conditions. Upon derivation and substitution, one would get an equation for y' and by integrating this rate of change, we could find the equation that governs y. Similarly, knowing y, we would back-substitute to find x. However, from the information provided, the steps for this specific solution process are not clear.
It's important to note that the reference equations provided, like (2.0 × 107+y) y = 1×10⁻¹⁴ or y² = v² + 2a(y − y₁), seem irrelevant to the task given. Therefore, without the correct differential equations corresponding to the system x²=4y and y²=2x, the solution cannot be found based on the provided references.