I think the original form works, but you just get eigenfunctions of sin(beta*L) = 0...see below...after doing it by hand, it appears that some of the reasoning might be too complicated for Maple to work out, plus there are some implicit assumptions used...
#Y_BCs:=y(0)=0, (D@@2)(y)(0)=0, y(L)=0, (D@@2)(y)(L)=0;
#Maples answer is messy, and convert doesn't give what I want so use form below
#BC at x=0
#so B = 0 and D = 0
#BC at x=L
#So C = 0 and eigenvalues are sin(beta*L) = 0