18.8 Mathematical methods and models in economics

Problem: In simple market models, supply and demand are usually supposed to depend only on the price of the product. In real situations, supply and demand also depend on the pricing trend and the rate of price change. In models with continuous and time-differentiable \( t \) functions, these characteristics are described by the first and second derivatives of the cost function \( p(t) \), respectively. Let the function of demand \( D \) and supply \( S \) have the following dependances on the post \( p \) and its derivatives: \( D(t)=p^{\prime \prime}-2 p^{\prime}-6 p+36, S(t)=2 p^{\prime \prime}+ \) \( +2 p^{\prime}+4 p+6 \). Find the dynamics of the equilibrium \( p \) price for the product dependent on time.