Hi,
this works as expected:
sols = NDSolve[{x[t] y'[t] + x'[t] y[t] == (x[t]^2) y[t] + (y[t]^2) x[t]; (x[t]^2) y'[t] + (y[t]^3) x'[t] == x[t] y[t] + (x[t]^2) y[t] + (y[t]^2) x'[t], x[0] == 0.1,
y[t] == 0.1}, {x, y}, {t, 0, 1.}]
Plot it
Plot[{x[t] /. sols, y[t] /. sols}, {t, 0, 0.5}]
and you get:

Cheers,
Marco