Final answer:
The method of separation of variables is used to solve the given partial differential equation. X(x) satisfies the differential equation X'' - μX = 0, while T(t) satisfies the corresponding differential equation obtained by dividing both sides by D. X(x) must satisfy the boundary conditions ∂x/∂θ = 0 when x = 0 for t > 0 and θ = 0 when x = L for t > 0. The non-trivial solutions of the differential equation satisfying the boundary conditions are X(x) = Acos(kx) and X(x) = Bsin(kx), where k can take any positive value.
Step-by-step explanation:
To solve the differential equation using the method of separation of variables, we assume that the temperature distribution can be expressed as a product of two functions: θ(x,t) = X(x)T(t). Substituting this into the given equation, we get X'' - μX = 0, where μ is a constant. The corresponding differential equation for T(t) is found by dividing both sides of the equation by D.
The two boundary conditions that X(x) must satisfy are ∂x/∂θ = 0 when x = 0 for t > 0 and θ = 0 when x = L for t > 0.
For the case where μ < 0, we have μ = -k^2 for some k > 0. The general solution of equation (*) is X(x) = Acos(kx) + Bsin(kx). The non-trivial solutions that satisfy the boundary conditions are X(x) = Acos(kx) and X(x) = Bsin(kx), where k can take any positive value.
The function f(x,t) = exp(-Dk^2t)cos(kx) satisfies the given partial differential equation for any constant k.