First separate of variable $$u(x,y) =X(x)Y(y),$$
take derivative
$$ X''Y + XY'' = -\lambda XY$$
$$\implies \frac{X''}{X}+\frac{Y''}{Y}= -\lambda $$
By the given condition
$$ X''+\lambda_1X = 0 , X'(0)+X'(a) = 0 $$
This is Neumann boundary condition
As n = 0,$$ \lambda_1 = 0 ,X_0 = \frac{1}{2}$$
As n = 1,2,...,$$ \lambda_1 = \frac{n^2\pi ^2}{a^2},X_1 = \cos\frac{n\pi x}{a}$$
For Y, this is Dirichlet boundary condition $$ Y''+\lambda_2Y= 0 , Y(0)+Y(b) = 0 $$
As m = 1,2,...,$$ \lambda_2 = \frac{m^2\pi ^2}{b^2},Y_1 = \sin\frac{m\pi y}{b}$$
$\lambda =\lambda_1 + \lambda_2 $, for n,m = 1,2,....
$$\implies \lambda = \pi^2(\frac{m^2}{b^2}+\frac{n^2}{a^2}) $$
for n,m = 1,2,....
$$ u(x,y) =\cos\frac{n\pi x}{a} \sin\frac{m\pi y}{b}$$
For n = 0, $\lambda$ = 0 and u(x,y) = 0
