Solving Second Order Linear Differential Equations in Economics

Published on June 4, 2026

Calculus

Learn how to solve a first-order linear differential equation derived from a beef market model to analyze price dynamics and market stability.

Question

please answer

Answer

Based on the provided image, here is the step-by-step solution to the problem.

Given:

The demand function

and supply function

for the beef market are:

a. Formulate the differential equation for the price in equilibrium

In equilibrium, the quantity demanded equals the quantity supplied:

Subtract

from both sides and rearrange the terms to one side:

Divide the entire equation by

to simplify:

Or in standard form:

(Note: Although the title says “Second Order”, the terms cancel out, resulting in a first-order linear differential equation).

b. Find the price time path

1. Complementary Function :
Solve the homogeneous equation

.
The characteristic equation is

.
Thus, the complementary solution is:

2. Particular Solution :
Since the right side is a constant, we assume

. Then

.
Substitute into the differential equation:

So,

General Solution:

c. Definitize the time path given and

(Note: Since the resulting equation is first-order, only one initial condition is needed to find . The condition is likely a distractor or based on the original second-order assumption before cancellation).

Using

The definite time path is:

d. Analyze the stability of this market

A market is stable if the price converges to its equilibrium value as

.
Looking at the solution:

Since the exponential term

decays to zero (because the coefficient of

is negative), the price

converges to the equilibrium price of

Conclusion: The market is stable.

🚀 Try Free Math Help Now!🏠 Learn More Powerful Features