Question

Solve the following differential equation:

Suppose that the quantity demanded Q_d = `13 - 6"p" + 2 "dp"/"dt" + ("d"^2"p")/("dt"^2)` and quantity supplied Q_s = `- 3 + 2"p"` where p is the price. Find the equilibrium price for market clearance

Answer 1

For market clearance, the required condition is Q_d = Q_s

`13 - 6"p" + 2 "dp"/"dt" + ("d"^2"p")/("dt"^2) = - 3 + 2"p"`

`("d"^2"p")/("dt"^2) + 2 "dp"/"dt" - 6"p" + 13 + 3 = 0`

`("d"^2"p")/("dt"^2) + 2 "dp"/"dt" - 8"p" = - 16`

`("D"^2 + 2"D" - 8)"p" = - 16`

The auxiliary equation is

m² + 2m – 8 = 0

(m + 4)(m – 2) = 0

m = – 4, 2

Roots are real and different

C.F = Ae^m₁t + Be^m₂t

C.F = `"Ae"^(- 4"t") + "BE"^(2"t")`

P.I = `1/(("D"^2 + 2"D" - 8)) (- 16)`

= `1/(("D"^2 + 2"D" - 8)) (- 16 "e"^(0_x))`

= `1/((0 + 2(0) - 8)) (- 16 "e"^(0_x))`

= `(-16)/(- 8)`

P.I = 2

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

P = `"Ae"^(-4"t") + "Be"^(2"t") + 2`