and

The solution to these differential equations can be written in the form of:

And

where e represents a variable generally called eccentricity (but which has a different meaning than the astronomical meaning of the term).

One can see that if the forcing frequency were exactly the same as the natural frequency, the denominator goes to zero and the amplitude of the oscillatory motion therefore goes to infinity. In mechanical systems, this is akin to the situation when a device disintegrates due to unexpected vibrations. These equations are for the situation for a system which has no damping factor, which cannot actually occur in any real mechanical device, but which is essentially true for systems of planets orbiting the Sun.

The claim here is that this Engineering approach of forced vibration can be applied for the situation of one planet perturbing another. The equations are more complex than these simple ones because the strength of the perturbing force constantly varies with the distance between the two planets per the inverse square rule, and they are each travelling in elliptic orbits. So the actual mathematics of this is more complex, but the reasoning is as indicated here.

Therefore, this shows that if the perturbing (forcing) frequency were commensurate with the natural frequency (inverse of period) of a planet, the perturbing effect would be extremely unstable and essentially catastrophic. This shows the well known effect of this fact that exact commensurability cannot exist among planets or satellites or asteroids with Jupiter.

There are therefore two adjacent orbital radii of near commensurability, one on each side of the exact commensurable orbital radius, where a meta-stable relationship can and will occur. By inserting the actual parameters for any specific two planets in the differential equations above, it should be possible to solve them to calculate the actual meta-stable solutions. Such solutions contain the above and also factors that depend on the cosine or sine of the motion of the perturbed planet, so the perturbative effects vary by both orbital motions. It is my belief that this should then indicate/calculate the slight differences of the Galilean satellites from exact commensurability with each other, and the other slight differences which exist in such relationships.