To find the equilibrium price where demand equals supply, we start with the given demand function:

$q_D = 12 - p - 3\frac{dp}{dt} + \frac{d^2p}{dt^2}$

and the supply function:

$q_S = -8 + 3p.$

Condition for Equilibrium

At equilibrium, demand equals supply:

$q_D = q_S.$

Substituting the functions into the equation gives:

$12 - p - 3\frac{dp}{dt} + \frac{d^2p}{dt^2} = -8 + 3p.$

Rearranging

Bringing all terms to one side, we have:

$\frac{d^2p}{dt^2} - 3\frac{dp}{dt} + 4p - 20 = 0.$

Solving the Differential Equation

This is a second-order linear homogeneous differential equation. The characteristic equation can be formed as follows:

$r^2 - 3r + 4 = 0.$

Finding Roots

To find the roots $r$, we use the quadratic formula:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 - 16}}{2} = \frac{3 \pm \sqrt{-7}}{2}.$

This gives complex roots:

$r = \frac{3}{2} \pm \frac{\sqrt{7}}{2} i.$

General Solution

The general solution to the differential equation is:

$p(t) = e^{\frac{3}{2}t}\left(A \cos\left(\frac{\sqrt{7}}{2} t\right) + B \sin\left(\frac{\sqrt{7}}{2} t\right)\right),$

where $A$ and $B$ are constants determined by initial conditions.

Applying Initial Conditions

Given:

1. $p(0) = 7$
2. $\frac{dp}{dt}(0) = 9$

Step 1: At $t = 0$

$p(0) = A = 7.$

Step 2: First Derivative

To find $B$, we differentiate $p(t)$:

$\frac{dp}{dt} = e^{\frac{3}{2}t}\left( \frac{3}{2}(A \cos(...)+ B \sin(...)) + e^{-\frac{3}{2}t}\left(-A \frac{\sqrt{7}}{2} \sin(...) + B \frac{\sqrt{7}}{2} \cos(...)\right)\right).$

At $t = 0$: $\frac{dp}{dt}(0) = \frac{3}{2}A + B\frac{\sqrt{7}}{2} = 9.$ Substituting $A = 7$: $\frac{3}{2}(7) + B\frac{\sqrt{7}}{2} = 9.$ $\frac{21}{2} + \frac{\sqrt{7}}{2}B = 9 \implies \frac{\sqrt{7}}{2}B = 9 - \frac{21}{2} = \frac{18 - 21}{2} = -\frac{3}{2}.$ Thus, $B = -\frac{3}{\sqrt{7}}.$

Final Solution

The equilibrium price as a function of time is:

$p(t) = e^{\frac{3}{2}t}\left(7 \cos\left(\frac{\sqrt{7}}{2} t\right) - \frac{3}{\sqrt{7}} \sin\left(\frac{\sqrt{7}}{2} t\right)\right).$

This represents the equilibrium price $p(t)$ over time given the initial conditions.

Questions or Clarifications?

Would you like to delve deeper into any part of this solution? Here are some follow-up questions:

1. Would you like to explore the implications of this equilibrium price on market behavior?
2. Are you interested in the stability of the equilibrium?
3. Would you like to see a graphical representation of the price function over time?
4. Do you want to analyze how changes in initial conditions affect the equilibrium?
5. Would you like to discuss applications of such models in real-world economics?

Tip: When working with differential equations, pay attention to initial conditions—they are key to finding specific solutions!