= Solution
Let $z=\omega_1^2$, so $\omega_2^2=4z$. The sum and product of roots of the <characteristic polynomial> in (i) give
$$
5z=\frac{2(m_2+1)}{m_2},\qquad 4z^2=\frac3{m_2}.
$$
Eliminating $z$ yields
$$
16(m_2+1)^2=75m_2,\qquad16m_2^2-43m_2+16=0.
$$
Therefore the <octave tuning in a two-mass complete spring graph> has two positive solutions:
$$
\boxed{m_2=\frac{43\pm5\sqrt{33}}{32}\simeq0.446162086\text{ or }2.241337914.}
$$
The two values are reciprocals. Both meet the requested <frequency ratio>; substituting either into the expressions in (i) gives $\omega_2/\omega_1=2$.
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