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CSS Applied Mathematics QUESTION #4160
Question 1
For the heat equation $u_t = k u_{xx}$ on a rod of length $L$ with insulated ends ($u_x(0,t) = u_x(L,t) = 0$), what are the eigenfunctions?
  • $\sin(\frac{n\pi x}{L})$
  • $\cos(\frac{n\pi x}{L})$✔️
  • $e^{\frac{n\pi x}{L}}$
  • $\sin(\frac{(2n+1)\pi x}{2L})$
Correct Answer Explanation
Insulated ends (Neumann BCs) $u_x = 0$ at $x = 0, L$ give eigenfunctions $\cos(\frac{n\pi x}{L})$ for $n = 0, 1, 2, ...$