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You may be better off with a numerical solution.

var('t x y')
F=[2*(-x)/sqrt(x^2+(t-y)^2),2*(t-y)/sqrt(x^2+(t-y)^2)]
ics=[3,10]
times=srange(0,7,0.5)
soln=desolve_odeint(F,ics,times,[x,y])
line(soln)