Solve the partial differential equation (d u)/(d t) = (d2 u)/(d x2) with boundary conditions u(0, t) = 0 and u(1, t) = 0 using the method of separation of variables.

Step 1: Separate variables. Assume a solution of the form u(x, t) = X(x)T(t). Substitute this into the given partial differential equation ( u)/( t) = (^2 u)/( x^2): X(x)T'(t) = X''(x)T(t) Divide by X(x)T(t) to separate the variables: (T'(t))/(T(t)) = (X''(x))/(X(x)) Since the left side depends only on t and the right side depends only on x, both sides must be equal to a constant, say -. (T'(t))/(T(t)) = - and (X''(x))/(X(x)) = - This gives two ordinary differential equations: 1. T'(t) + T(t) = 0 2. X''(x) + X(x) = 0 Step 2: Solve the spatial equation X''(x) + X(x) = 0 using the boundary conditions. The boundary conditions are u(0, t) = 0 and u(1, t) = 0. From u(0, t) = X(0)T(t) = 0, we get X(0) = 0 (assuming T(t) is not identically zero). From u(1, t) = X(1)T(t) = 0, we get X(1) = 0. Consider the possible values for : If = 0: X''(x) = 0 X(x) = Ax + B. X(0) = 0 B = 0. X(1) = 0 A = 0. This gives X(x) = 0, a trivial solution. If < 0: Let = -^2 for > 0. X''(x) - ^2 X(x) = 0. The general solution is X(x) = A e^ x + B e^- x. X(0) = 0 A + B = 0 B = -A. X(x) = A(e^ x - e^- x) = 2A ( x). X(1) = 0 2A () = 0. Since > 0, () ≠ 0, so A = 0. This again gives a trivial solution. If > 0: Let = ^2 for > 0. X''(x) + ^2 X(x) = 0. The general solution is X(x) = A ( x) + B ( x). X(0) = 0 A (0) + B (0) = 0 A = 0. So, X(x) = B ( x). X(1) = 0 B () = 0. For a non-trivial solution, B ≠ 0, so () = 0. This implies = n for n = 1, 2, 3, . The eigenvalues are _n = (n )^2. The eigenfunctions are X_n(x) = (n x) (we absorb the constant B into the general solution coefficients). Step 3: Solve the temporal equation T'(t) + T(t) = 0. For each eigenvalue _n = (n )^2, the equation becomes T'(t) + (n )^2 T(t) = 0. This is a first-order linear ODE. (dT)/(dt) = -(n )^2 T Integrating both sides: (dT)/(T) = -(n )^2 dt |T| = -(n )^2 t + C' T_n(t) = C_n e^-(n )^2 t Step 4: Form the general solution. The general solution is a superposition of all possible solutions: u(x, t) = _n=1^ X_n(x)T_n