DSolve does nothing on ODE equation system

Although DSolve indeed cannot solve the system of ODEs in the question, it can solve the equivalent system,

DSolve[{z'[t] == (2 z[t] y'[t])/y[t], y''[t] == (-z[t]^2 + y'[t]^2)/y[t]}, {z[t], y[t]}, t]
    // Simplify

where z[t] is D[x[t], t].

(* {{y[t] -> -((I Sqrt[C[2]] Coth[Sqrt[C[2]] (t - C[3])] 
         Sqrt[Sech[Sqrt[C[2]] (t - C[3])]^2])/C[1]), 
     z[t] -> -((C[2] Csch[Sqrt[C[2]] (t - C[3])]^2)/C[1]), 
    {y[t] -> -((I Sqrt[C[2]] Coth[Sqrt[C[2]] (t - C[3])] 
         Sqrt[Sech[Sqrt[C[2]] (t - C[3])]^2])/C[1]), 
     z[t] -> -((C[2] Csch[Sqrt[C[2]] (t - C[3])]^2)/C[1]), 
    {y[t] -> y[t] -> (I Sqrt[C[2]] Coth[Sqrt[C[2]] (t - C[3])] 
         Sqrt[Sech[Sqrt[C[2]] (t - C[3])]^2])/C[1], 
     z[t] -> -((C[2] Csch[Sqrt[C[2]] (t - C[3])]^2)/C[1]), 
    {y[t] -> (I Sqrt[C[2]] Coth[Sqrt[C[2]] (t - C[3])] 
         Sqrt[Sech[Sqrt[C[2]] (t - C[3])]^2])/C[1], 
     z[t] -> -((C[2] Csch[Sqrt[C[2]] (t - C[3])]^2)/C[1])}} *)

An additional integration then gives x[t]. There are four solutions, because the ODEs are nonlinear in both x and y.

It is surprising how often DSolve is unable to solve relatively simple systems of ODEs.

I also get no solution from DSolve in version 11.0.1. But if it any help, here is a solution I borrowed from Maple, which was able to solve it,in Mathematica code, that you could try. (Hopefully future version of Mathematica will also be able to solve this).

The solution contains 4 constants of integrations, since these are two second order ODE's. There are more solutions given. You would have to now solve for these 4 constants yourself (called them C[1],C[2],... etc.. using different initial conditions ofcourse in order to plot these solutions.

x[t_] := C[4] - 2*Tanh[t/(2*C[1]) + C[3]/(2*C[1])];
y[t_]:=(2 Sech[t/(2 C[1]) + C[3]/(2 C[1])]^4)/(C[1]^2 Sqrt[(
 2 Sech[t/(2 C[1]) + C[3]/(2 C[1])]^4 Tanh[
t/(2 C[1]) + C[3]/(2 C[1])]^2)/C[1]^4 + (
 2 Sech[t/(2 C[1]) + C[3]/(
 2 C[1])]^2 (Sech[t/(2 C[1]) + C[3]/(2 C[1])]^4/(2 C[1]^3) - (
 Sech[t/(2 C[1]) + C[3]/(2 C[1])]^2 Tanh[
   t/(2 C[1]) + C[3]/(2 C[1])]^2)/C[1]^3))/C[1]])

Here is the screen shot showing the other solutions in case you need them