Solving a Logistic Differential Equation

1Introduction

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.

2The Equation

We are given the following logistic differential equation:

$$\frac{dN}{dt} = 9 \cdot 10^{-6} \cdot N(7000 - N)$$

Initial condition:

$$N(0) = 600$$

3Step 1: Separation of Variables

Separate variables:

$$\frac{1}{N(7000 - N)} \, dN = 9 \cdot 10^{-6} \, dt$$

4Step 2: Partial Fraction Decomposition

We split the left-hand side:

$$\frac{1}{N(7000 - N)} = \frac{A}{N} + \frac{B}{7000 - N}$$

Multiply through:

$$1 = A(7000 - N) + BN$$

Solve for $A$ and $B$:

• Let $N = 0 \Rightarrow A = \frac{1}{7000}$
• Let $N = 7000 \Rightarrow B = \frac{1}{7000}$

So:

$$\frac{1}{N(7000 - N)} = \frac{1}{7000} \left( \frac{1}{N} + \frac{1}{7000 - N} \right)$$

5Step 3: Integration

$$\frac{1}{7000} \int \left( \frac{1}{N} + \frac{1}{7000 - N} \right) dN = \int 9 \cdot 10^{-6} dt$$

Integrate:

$$\frac{1}{7000} \left( \ln|N| - \ln|7000 - N| \right) = 9 \cdot 10^{-6} t + C$$

$$\ln \left| \frac{N}{7000 - N} \right| = 0.063t + C$$

6Step 4: Solve for $N(t)$

Exponentiate both sides:

$$\frac{N}{7000 - N} = e^{0.063t + C} = Ke^{0.063t}$$

Solve for $N$:

$$N = \frac{7000 \cdot Ke^{0.063t}}{1 + Ke^{0.063t}}$$

7Step 5: Apply Initial Condition

Given: $N(0) = 600$

$$600 = \frac{7000K}{1 + K} \Rightarrow 600 + 600K = 7000K \Rightarrow 600 = 6400K \Rightarrow K = \frac{3}{32}$$

8Final Solution

$$N(t) = \frac{7000 \cdot \frac{3}{32} e^{0.063t}}{1 + \frac{3}{32} e^{0.063t}}$$

9Step 6: Approximate Exponential (Optional)

The approximation $e^{0.063t} \approx 1.065^t$ helps with calculator-free computation.

Using the approximation:

$$e^{0.063t} \approx 1.065^t$$

Substitute:

$$N(t) \approx \frac{7000 \cdot \frac{3}{32} \cdot 1.065^t}{1 + \frac{3}{32} \cdot 1.065^t}$$

Multiply numerator and denominator by $\frac{32}{3}$:

$$N(t) \approx \frac{7000 \cdot 1.065^t}{\frac{32}{3} + 1.065^t} = \frac{7000 \cdot 1.065^t}{10.667 + 1.065^t}$$