Substituting in the budget constraint to find the individual demand function:
\(P_X+P_Y=P_X X+ \frac{P_X}{2} X\)
\(X= \frac{2}{3}+\frac{2}{3}\frac{P_Y}{P_X}\)
And:
\(P_X+P_Y=P_Y Y+ 2P_Y Y\)