Question

Suppose that Q_d = `30 - 5"P" + 2 "dP"/"dt" + ("d"^2"P")/("dt"^2)` and Q_s = 6 + 3P. Find the equilibrium price for market clearance

Answer 1

Q_d = `30 - 5"P" + 2 "dP"/"dt" + ("d"^2"P")/("dt"^2)` and Q_s = 6 + 3P

For market clearance, the required condition is

Q_d = Q_s 

`30 - 5"P" + 2 "dP"/"dt" + ("d"^2"P")/("dt"^2)` = 6 + 3P

`("d"^2"P")/("dt"^2) + 2 "dP"/"dt" - 5"P" - 3"P" + 30 - 6` = 0

`("d"^2"P")/("dt"^2) + 2 "dP"/"dt" - 8"P" + 24` = 0

`("D"^2 + 2"D" - 8)"P" = - 24`

The auxiliary equation is m2 + 2m – 8 = 0

(m + 4)(m – 2) = 0

m = -4, 2

Roots are real and different

C.F = Ae^m₁x + Be^m₂x

C.F = Ae^-4t + Be^2t 

P.I = `1/(("D"^2 + 2"D" - 8)) (- 24)`

= `1/(("D"^2 + 2"D" - 8)) (-24"e"^(0_x))`

=  `(-24"e"^(0x))/(0 + 2(0) - 8)`

=  `(-24)/(-8)`

= 3

The general solution is y = C.F + P.I

y = Ae^-ut + Be^2t + 3