Question

if the supply and the demand of a certain product is given by
D(t)=95-5p(t)+2p'(t) and
S(t)=35-p(t)+3p'(t) respectively where p(t) is the price at any time t. Find the equilibrium price and the long range equilibrium price given that p(0)=50

Answer 1

$$D(t)=S(t)$$

$$95-5p(t)+2p'(t)=35-p(t)+3p'(t)$$

$$p'(t) =-4p(t)+60$$

$$\dfrac{dp}{15-p}=4dt$$

Integrate

$$\int\dfrac{dp}{15-p}=\int 4dt$$

$$\ln|15-p|=-4t+\ln C$$

$$|15-p|=Ce^{-4t}$$

$$p(0)=50, |15-50|=C$$

$$C=35$$

The equilibrium price 

$$p(t)=15+35e^{-4t}$$

The long range equilibrium price 

$$p=15$$