Suppose that consumer demand depends upon the price-trend according to the formula [8]
q = 12 − p − 3
dp
dt
+
d2p
dt2 .
If the supply function qS(p) = −8 + 3p, write down the condition for equilibrium and determine the
equilibrium price p(t) when p(0) = 7 and p′(0) = 9

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!

Suppose that consumer demand depends upon the price-trend according to the formula q = 12 − p − 3(dp/dt) + (d^2p/dt^2). If the supply function qS(p) = −8 + 3p, write down the condition for equilibrium and determine the equilibrium price p(t) when p(0) = 7 and p′(0) = 9.

Explore how to determine the equilibrium price in a market where consumer demand and supply are defined by specific functions. This detailed analysis includes the equilibrium condition and a step-by-step solution for finding the price function over time, given initial conditions. Ideal for high school students in grades 11-12 studying economics and differential equations.

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