Just so there is less concealed, this is how I derived the pressure relationship:
I applied Newtons 2nd for Accelerating frame:
$$ \sum F - m \ddot x = m \ddot z $$
Where;
## \ddot x ## is the acceleration of the oscillator
## \ddot z ## is the acceleration of the fluid element in the accelerating frame
From here I chose to make some simplifying assumption that the area of the cup is much larger than the jet so that ## \ddot z ## can be neglected.
This implies that:
$$ \sum F = m \ddot x $$
$$ PA - ( P + dP )A + \rho A g dz = \rho A dz \frac{d^2 x}{dt^2} $$
This simplifies to:
$$ \frac{dP}{dz} = - \rho \left( \frac{d^2 x}{dt^2} - g \right) $$
Integrating that from the free surface of the water to the bottom of the cup gives the result:
$$ P = - \rho \left( \frac{d^2 x}{dt^2} - g \right) z $$
___________________________________________________________________________________________________________________
If you don't make this simplifying assumption above, you add a third ODE to the system, and it becomes even more hideous:
$$ \frac{dP}{dz} = - \rho \left( \frac{d^2 x}{dt^2} + \frac{d^2 z}{dt^2} - g \right) $$
Remembering:
$$ z = \frac{M(t) - m_b}{A_b} $$
It follows that:
$$ \frac{d^2 z}{dt^2} = \frac{1}{A_b} \frac{d^2 }{dt^2} \left( M(t) \right) $$
and
$$ \frac{d}{dz} \left( M(t) \right) = A_b $$
I find that the resulting equation is given by:
$$ A_b \frac{dP}{dt} = - \rho \left( \frac{d^2 x}{dt^2} + \frac{1}{A_b} \frac{d^2 }{dt^2} \left( M(t) \right) - g \right) \frac{d}{dt} ( M(t) ) $$