I wish we had the dot notation! You will find that, even if it does not offer everything, this site offers much more then most in the way of notation.
To your question, let me use a prime (') in place of the dot
x'(t) = C
Note that I added the implied variable, now we need to evaluate the constant for all time, we have the initial condition of
But since x'(t)=C we must have C=v0
repeat this logic to get evaluate your constant of integration for the y equation.
Simply solve your equation for x(t) for t, then take the resulting function of x and subsitiute it into your y(t) equation.
In the future you may wish to post such questions to our Homework help fourm.