Let u be the solution to the initial boundary value problem for the Heat Equation, du(t, x) = 50/u(t, x), t≤ (0, ∞), x = (0,3); with Neumann boundary conditions du(t,0) = 0 and du(t, 3) = 0 and with initial condition u(0, x) = f(x) = Co = Cn = u(t, x) The solution u of the problem above, with the conventions given in class, has the form = with the normalization conditions v₁ (0) = 1 and wn (0) vn (t) = wn(x) 3, 4, Co 90 +29 2 n= x = 0, x E Cn Un (t) wn(x), = 1. Find the functions Un, wn, and the constants co and cn for n > 1. Σ M M Σ

Verify that the given functions y1(t)=t, y2(t)=t?^t are solutions to the homogeneous equation corresponding to t^2y″−t(t+2)y′+(t+2)y=2t^3, t ∈(0, infinity) then find a particular solution of the nonhomogeneous equation.

Determine the particular solution z(x, y) of the partial differential equation a²z 10x³y + 12x?y2 ax2 subject to the boundary conditions z = y and az = y? ax when x = 1.

(10) Using u(x,t) = (A(p) cos pr + B(p) sin pre-p² dp obtain the integral form solution of Ət² = , 2u მე2 satisfying the initial condition u(x, 0) = f(x), where f(x) = 1