1) Let ­ = [0; a] £ [0; b] ½ R2. We seek the (unique) solution u : ­ ! R to the Laplace boundary-value problem ¢u = 0 in ­; u(0; y) = u(x; 0) = 0; for 0 · x · a; 0 · y · b; ux(a; y) = 0; for 0 · y · b; u(x; b) = g(x); for 0 · x · a; where g : [0; a] ! R is a known function. Using the method of separation of variables, calculate the solution up to unknown constants, and give a condition that can enable us to calculate these unknown constants.