As the link you provided mentions:
"Differential equations can be used to include the dynamic aspects to
economics into a mathematical framework which takes into account the
volatility present in economics."
So the author does not derive the differential equations as the result of a theoretical model -the author just wants to obtain the movement through time of equilibrium price (the author assumes market-clearing), as a second order differential equation.
So it is a purely descriptive/forecasting exercise, devoid of any economic theory background. The author obtains the solution, which includes a number of coefficients to be determined, for which the author mentions
"These values can be estimated using statistics and econometric
Some constraints on the parameters may come from observing the past price process (is it oscillating? around a trend or around an horizontal line? etc), and in the end, the hope is that we may obtain a good predictor for the price movements -again, without basing it on economic theory.