= Solution
Set $\omega_1=u$, $\omega_2=\Omega+v$, $\omega_3=w$ and retain terms linear in the perturbations in <Euler equations for a torque-free rigid body>. Using $I_3=I_1+I_2$ gives
$$
\dot u=-\Omega w,\qquad\dot v=0,\qquad
\dot w=-\frac{a^2-b^2}{a^2+b^2}\Omega u.
$$
Consequently $\ddot u=\beta^2\Omega^2u$, where $\beta=\sqrt{(a^2-b^2)/(a^2+b^2)}>0$. For $\Omega>0$ the growing solution is
$$
\boxed{u=C e^{\beta\Omega t},\qquad w=-\beta C e^{\beta\Omega t}.}
$$
There is also a decaying mode with the opposite exponent. For negative $\Omega$, select the exponent $+\beta|\Omega|$ as the growing one. This is the <intermediate axis theorem>: steady rotation about the intermediate principal inertia axis is unstable whenever $\Omega\ne0$.
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