5. (3 points) Assume on some (smooth, bounded) domain $\Omega \subset \mathbb{R}^2$ the solution of the heat equation
$$ \frac{\partial u}{\partial t} - \Delta u = 0, \quad (x, t) \in \Omega \times [0, \infty), \quad u(x, t) = 0 \quad (x, t) \in \partial \Omega \times [0, \infty) $$
has the form
$$ u(x, t) = \sum_{k=1}^{\infty} A_k e^{-\lambda_k^2 t} \phi_k(x) $$
for some functions $\phi_k$ and numbers $\lambda_k, A_k, k = 1, 2, \dots$. Then the solution of the wave equation
$$ \frac{\partial^2 v}{\partial t^2} - \Delta v = 0, \quad (x, t) \in \Omega \times [0, \infty), \quad v(x, t) = 0 \quad (x, t) \in \partial \Omega \times [0, \infty) $$
has the form
$$ v(x, t) = \sum_{k=1}^{\infty} (B_k \cos(\lambda_k t) + C_k \sin(\lambda_k t)) \phi_k(x) $$
for the same functions $\phi_k$ and numbers $\lambda_k$, but possibly different $B_k$ and $C_k$.
$\bigcirc$ True, the fundamental modes $\phi_k$ are the same and the algebra works out.
$\bigcirc$ False, we need to replace $\cos(\lambda_k x)$ and $\sin(\lambda_k x)$ respectively by $\cos(\sqrt{\lambda_k} x)$ and $\sin(\sqrt{\lambda_k} x)$.
$\bigcirc$ False, the fundamental modes $\phi_k$ are different for the heat and wave equation.
$\bigcirc$ False, the numbers $B_k$ and $C_k$ should be equal to $A_k$.