Background on Differential Equations:
“It is a truism that nothing is permanent except change; and the primary purpose of differential equations is to serve as a tool for the study of change in the physical world…these applications include:
The brachistochrone problem
The Einstein formula e=mc*c
Newton’s law of gravitation
The wave equation for the vibrating spring
The harmonic oscillator in quantum mechanics
The wave equation for a vibrating membrane
The predator-prey equations
Abel’s mechanical problem
I consider the mathematical treatment of these problems to be among the chief glories of Western civilization…” (Simmons, 91)
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. The propagation of waves across a medium can be adopted in economics in various forms including structural shocks to the economy, diffusion of monetary stimulus throughout various sectors in the economy and the interaction between producer and consumer when considering current prices and their derivatives. Linear as well as Non-Linear Differential equations have their place in economics, but there is still much work to be done. Chaos Theory and Non-Linear Differential equations are on the cutting edge of current economic research.
We need to write the equations for supply and demand in terms of price (P), the rate of change of the price (P’), and the rate of change of the rate of change of the price (P’). The values given to w, u, and v depends on the peoples expectations about how prices are changing. If people think that prices are rising then the coefficient in front of the first derivative of price will be positive and if there is a belief that prices are falling then this coefficient will be negative. The magnitude and since of the v value reflects how fast people believe that prices are rising or falling. These values can be estimated using statistics and econometric methods, but the following solution is for the general case where these variable are arbitrary real numbers not equal to zero.
Assuming that quantity supplied is equal to quantity demanded, which is the typical assumption in equilibrium models, and which might seem harmless, but in this assumption has many critics and it is worth noting that these criticisms have powerful logic arguments and should not be ignored. For now, this is the assumption that will be used, since the purpose is to explore the dynamic path of prices as they related to current conditions in an equilibrium setting.
We can make a substitution for the difference of constants in front of the price level and its derivatives and divide by the coefficient in front of the second derivative to get the equilibrium equations in the standard form.
To solve this differential equation we must solve for the particular and the homogeneous solutions by recalling the formula for the particular solution when we have a constant and finding the characteristic roots of the characteristic equation, once we have the characteristic roots using the quadratic formula we need to use the discriminant to break out general solution into 3 categories: two real distinct solutions, one real repeated solution, and two complex conjugates.
The general solution to the differential equation with two real and distinct roots is :
The general solution to the differential equation with one real repeated root is:
The general solution to the differential equation with two complex conjugates is:
Reference: Schaum’s Outlines-Introduction to Mathematical Economics