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I came across this discussion, claiming supply and demand are expressible in terms of its price as $$Q_s=c_1+w_1P+u_1P'+v_1P'',\,Q_d=c_2+w_2P+u_2P'+v_2P''.$$Equating supply to demand for an equilibrium gives an ODE one can solve, but where do these formulae for $Q_s,\,Q_d$ come from?

I apologise if, contrary to the impression the page gives, these equations are not a mainstream model.

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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 methods"

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.

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