Don't put your constants inside brackets. Also, Dsolve doesn't take a starting and ending value for the dependent variable.
In[5]:= Nutrient = {x'[t] == (k/Va) (y[t] - x[t]),
y'[t] == (k/Vb) (x[t] - y[t]), x[0] == x0, y[0] == y0};
In[6]:= Solution = DSolve[Nutrient, {x[t], y[t]}, {t}]
Out[6]= {{x[t] -> (
Va x0 + E^((k t (-Va - Vb))/(Va Vb)) Vb x0 + Vb y0 -
E^((k t (-Va - Vb))/(Va Vb)) Vb y0)/(Va + Vb),
y[t] -> -((-Va x0 + E^((k t (-Va - Vb))/(Va Vb)) Va x0 -
E^((k t (-Va - Vb))/(Va Vb)) Va y0 - Vb y0)/(Va + Vb))}}