You can see that any multiple of a solution is also a solution. Unless the only solution is zero, the problem has infinitely many solutions, and NDSolve is not appropriate for the task. One nonzero solution is found easily: y[t_] = t^2, x[t_] = 2 t^2 for t>0. Here is a derivation that uses Mathematica.

The equation y[2 t] == 2 x[t] means that x == Function[t, y[2 t]/2]. Let us replace into the other equation:

2  y'[t] == x'[t] /. x -> Function[t, y[2 t]/2]

which gives

2 y'[t] == y'[2 t]

A sufficient condition for this relation is that y' is a monomial of the first degree, for example of the form 2 t. Integrate this and you get y[t_] = t^2. Go back and you find the corresponding x.