Solving differential equations is a fundamental skill in mathematics, science, and engineering. These equations describe how quantities change over time, making them essential for modeling real-world systems — from population growth and disease spread to physics simulations and financial predictions.
Understanding how to solve them analytically (as in this example) not only deepens mathematical intuition but also complements programming-based approaches. In many cases, numerical solvers in code (e.g., using Python or MATLAB) are used to approximate solutions, but knowing the underlying math helps debug, optimize, and better interpret those results.
This note walks through the full solution of a logistic growth model — a classic real-world example of a differential equation with limits on growth.
The Equation
We are given the following logistic differential equation:
Initial condition:
Step 1: Separation of Variables
Separate variables:
Step 2: Partial Fraction Decomposition
We split the left-hand side:
Multiply through:
Solve for and :
- Let
- Let
So:
Step 3: Integration
Integrate:
Step 4: Solve for
Exponentiate both sides:
Solve for :
Step 5: Apply Initial Condition
Given:
Final Solution
Step 6: Approximate Exponential (Optional)
The approximation helps with calculator-free computation.
Using the approximation:
Substitute:
Multiply numerator and denominator by :