Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 7 / b / ii / Solution

Let $z={\omega }_1^2$, so ${\omega }_2^2=4z$. The sum and product of roots of the characteristic polynomial in (i) giveEliminating $z$ yieldsTherefore the octave tuning in a two-mass complete spring graph has two positive solutions:The two values are reciprocals. Both meet the requested frequency ratio; substituting either into the expressions in (i) gives ${\omega }_2/{\omega }_1=2$.