Consider equation $\frac{∂u}{∂t}−\frac{∂^2u}{∂x^2}= 0, 0 ≤ x ≤ l$,
subject to Neumann BCs
$u_x(0, t) = 0, u_x(l, t) = 0,$ and initial condition $u(x, 0) = f(x)$, where $f(x) = sin(x(l − x))$.
Show that $u(x, t) = u(l − x, t)$ for all $x ∈ [0, l]$ and all $t ≥ 0$.
I'm unsure where to start with this proof, however I know how to show that the solution is unique and think that this would be helpful when proving this.
Here's an idea to start. You know that $u(x,t)$ satisfies the heat equation, so perhaps see if you can differentiate $u(l-x,t)$ with respect to space and time and see if it looks like the heat equation. Then, as you said, you know something about uniqueness. So if you find that $u(l-x,t)$ solves the same problem as $u(x,t)$, what can you conclude?