I acknowledge similar questions have been answered, but despite this I still fail to see the solution to my problem.
Note that a,b, K are constants.
To the above PDE, I have by way of the product method $U(x,y) = X(x) Y(y)$ found an expression for $X$, but I am struggling with the second condition for $Y(y)$, specifically $U(x,b)=Y(b)=K\delta(x-x_{0})$
I have that $$Y(y)=Acosh(\frac{k\pi}{a}x) + Bsinh(\frac{k\pi}{a}x)$$ giving me after the first condition $Y(0)=0$, that $$Y(b)=Bsinh(\frac{k\pi}{a}b)=K\delta(x-x_{0})$$ which is where I fail to see where to proceed (assuming I haven't made some prior mistake).