= Solution
For independent $\chi,\chi^*$, the derivatives needed in the $\chi^*$ <Euler-Lagrange field equation> are
$$
\frac{\partial\mathcal L_2}{\partial\chi^*}=\frac{i\hbar}{2}\dot\chi-\mu(\chi+\chi^*),\qquad
\frac{\partial\mathcal L_2}{\partial\dot\chi^*}=-\frac{i\hbar}{2}\chi,\qquad
\frac{\partial\mathcal L_2}{\partial(\nabla\chi^*)}=-\frac{\hbar^2}{2M}\nabla\chi.
$$
Consequently the two conjugate field equations are
$$
\boxed{i\hbar\dot\chi=-\frac{\hbar^2}{2M}\nabla^2\chi+\mu(\chi+\chi^*),\qquad
-i\hbar\dot\chi^*=-\frac{\hbar^2}{2M}\nabla^2\chi^*+\mu(\chi+\chi^*).}
$$
Write $\chi=\chi_1+i\chi_2$ with real components, and put $A=\hbar^2/(2M)$. Separating real and imaginary parts gives
$$
-\hbar\dot\chi_2=(-A\nabla^2+2\mu)\chi_1,\qquad
\hbar\dot\chi_1=-A\nabla^2\chi_2.
$$
For each real field take the real part of a complex <plane wave> amplitude, $\chi_j=\operatorname{Re}[q_j e^{i(\mathbf k\cdot\mathbf r-\omega t)}]$. With $\epsilon_k=\hbar^2|\mathbf k|^2/(2M)$ the two equations become
$$
i\hbar\omega q_2=(\epsilon_k+2\mu)q_1,\qquad
-i\hbar\omega q_1=\epsilon_k q_2.
$$
A nonzero <normal mode> requires their <determinant> to vanish, so
$$
\hbar^2\omega^2=\epsilon_k(\epsilon_k+2\mu).
$$
The positive-frequency <Bogoliubov quasiparticle dispersion> is therefore
$$
\boxed{\omega(\mathbf k)=\frac1\hbar\sqrt{\epsilon_k(\epsilon_k+2\mu)}
=\sqrt{\frac{|\mathbf k|^2}{2M}\left(\frac{\hbar^2|\mathbf k|^2}{2M}+2\mu\right)}.}
$$
There is also the negative-frequency conjugate solution, as required for real component fields. Since $\mu>0$, $\omega^2\geq0$, consistent with the stability of the chosen uniform <ground state>.
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