Final answer:
To solve for u(x, y) given the conditions △u = 0 and boundary conditions, we can use the method of separation of variables. By separating the variables and solving the resulting ordinary differential equations, we obtain the general solution u(x, y) = ∑[A_n*sin(√(n^2π^2)*x)*sinh(n^2π(y-1))] where n = 1, 2, 3, ..., and A_n are constants.
Step-by-step explanation:
To solve for u(x, y) given the conditions △u = 0 and boundary conditions u(x, 0) = u(0, y) = u(x, 1) = 0, we can use the method of separation of variables. We assume that u(x, y) can be written as u(x, y) = X(x)Y(y), and substitute this into the Laplace's equation △u = 0. By separating the variables, we obtain two ordinary differential equations: X''(x)/X(x) = -λ and Y''(y)/Y(y) = λ, where λ is a constant.
Solving these equations gives us eigenvalues and eigenfunctions. The eigenvalue equation for X(x) has the general solution X(x) = A*cos(√(λ)*x) + B*sin(√(λ)*x), and the eigenvalue equation for Y(y) has the general solution Y(y) = C*cosh(√(λ)*(y-1)) + D*sinh(√(λ)*(y-1)). Applying the boundary conditions, we find that B = 0 and C = 0.
Finally, combining the solutions for X(x) and Y(y), we obtain the solution u(x, y) = ∑[A_n*sin(√(n^2π^2)*x)*sinh(n^2π(y-1))] where n = 1, 2, 3, ..., and A_n are constants.