  A system of two bilinearly coupled harmonic oscillators has been solved
analytically by using the Born-Oppenheimer (BO) product wavefunction ansatz and
the phase-space bound trajectory approach [J. S. Molano et al., Chem. Phys.
Lett. \textbf{76}(12), 138171 (2021)]. The bilinearly coupled oscillator system
allows to obtain the analytical expression of the quantum system, facilitating
comparison with the results of using the BO ansatz product. The analytical and
BO wavefunctions are obtained as a product of parabolic cylinder functions. The
arguments of the parabolic cylinder functions of the exact and the BO
wavefunctions are related to the physical coordinates by linear
transformations, represented by matrices $\mathsf{G}$ and $\mathsf{\tilde{G}}$
respectively. A $QU$ decomposition of the matrix $\mathsf{G}$ outputs an upper
triangular matrix $\mathsf{U}$ that is closely related to the BO
$\mathsf{\tilde{G}}$ matrix. The effect of the BO non-adiabatic coupling is
analyzed on the stability of the phase-space equations of motion. The
eigenvalues and eigenfunctions obtained by the Born-Oppenheimer phase-space
trajectory approach show excellent agreement with the analytical solutions.
