Modify Exercise 3.4.9 if instead ∂u/∂x(0, t) = 0 and ∂u/∂x(L, t) = 0. Reference Exerc...

Modify Exercise 3.4.9 if instead ∂u/∂x(0, t) = 0 and ∂u/∂x(L, t) = 0.

Reference Exercise 3.4.9:

Consider the heat equation with a known source q(x, t):

Assume that q(x, t) (for each t > 0) is a piecewise smooth function of x. Also assume that u and ∂u/∂x are continuous functions of x (for t > 0) and ∂²u/∂x² and ∂u/∂t are piecewise smooth. Thus,

Justify spatial term-by-term differentiation. What ordinary differential equation does b_n(t) satisfy? Do not solve this differential equation.