Discussion
Consider the boundary value problem :
\(u_{t}(x, t) = u_{xx}(x, t)\), \(0 \lt x \lt c\), \(t \gt 0\),
with conditions :
\(u_{x}(0, t) = 0 = u_{x}(c, t)\), \(t \gt 0\),
\(u(x, 0) = f(x)\), \(0 \lt x \lt c\).
Substituting \(u(x, t) = X(x)\,T(t)\) in the above equation we generate the following Strüm-Liouville problem for \(X(x)\) :
Answer: B
The problem is \(X'' + \lambda X = 0\), \(X'(0) = X'(c) = 0\).
Substituting \(u = X(x)T(t)\) into \(u_{t} = u_{xx}\) gives \(XT' = X''T\), so \(T'/T = X''/X = -\lambda\), a constant. This yields \(X'' + \lambda X = 0\).
The boundary conditions are on \(u_{x}\): \(u_{x}(0, t) = X'(0)T(t) = 0\) for all \(t\), and similarly at \(x = c\). For a non-trivial \(T\) this forces \(X'(0) = 0\) and \(X'(c) = 0\), Neumann conditions (insulated ends).
Option A has Dirichlet conditions, which would come from \(u = 0\) at the ends. Options C and D mix the two kinds or bring in the value 1, which no condition gives.
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