Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S7 / 2 / c / Solution

Substitution of the exponential ansatz gives the homogeneous amplitude systemA nonzero mode requires zero determinant. Thus $\boxed{({\omega }_1^2-{\omega }^2)({\omega }_2^2-{\omega }^2)=4{\Omega }^2{\omega }^2}$ with the stated two effective stiffnesses. The off-diagonal product is positive, which determines the sign on the right.