Hi, i attempted this question, but not sure whether it is correct or not..

using separation of variables:

X''(x)=S X(x) , Y''(y)=-S Y(y), where S is separation constant

becos X'(0)=0, X'(1)=0, so using the boundary value table i get:

S=(npi)^2

X=cos(n pi x) n=0,1,2....

Y=Acos*square root of S * y + Bsin* square root of S * y

since Y(0)=0

so Y=Bsin(n pi y)

so X(x)Y(y)=Bcos(n pi x)* sin(n pi y)

i know it's not finished, but is it correct from here? cos i dont feel quite sure about it....thank you~