The equation for Y is easily solved
In[36]:= DSolve[Y'[t] == Y[t]^2 + 3, Y[t], t] Out[36]= {{Y[t] -> Sqrt[3] Tan[Sqrt[3] t + Sqrt[3] C[1]]}}
Determine C[1] using the initial condition, say Y[0]. Then solve DE for X. You can then evaluate V.
