Decay constant

Differential equations describe laws of change themselves—not just a single number, but rather the function(s) that satisfy a given relationship governing how quantities evolve.

I will distinguish among the differential equation itself, initial conditions, and its solution. Without initial conditions, we typically obtain only a family of solutions.

In the previous article, we explored multivariable calculus and its applications to real-world problems. This article introduces a new topic: differential equations—a fundamental tool for modeling how functions change. They are essential for understanding diverse physical phenomena, social science models, and engineering problems.

What Is a Differential Equation?

A differential equation is an equation involving an unknown function and one or more of its derivatives. Our primary goal is usually to find that unknown function. Conceptually, it expresses a relationship about the rate of change of a function.

When grasping the foundational concepts of differential equations, focus first on:

• the unknown function,
• the order of derivatives involved,
• initial (or boundary) conditions,
• general versus particular solutions, and
• the underlying real-world process of change.

A simple example of a differential equation is:

$$\frac{dy}{dx} = f(x, y)$$

Here, $y$ is the unknown function, $x$ is the independent variable, and $f(x, y)$ is a known function.

Classification

Differential equations can be classified along several dimensions:

1. By the number of independent variables:

   • Ordinary differential equations (ODEs): involve only one independent variable. The example above is an ODE.
   • Partial differential equations (PDEs): involve two or more independent variables; a typical form is:
   $$\frac{\partial u}{\partial t} = f\!\left(t, x, u, \frac{\partial u}{\partial x}\right)$$

2. By order:

   • First-order differential equations: contain only first derivatives, e.g., $\frac{dy}{dx} = y$.
   • Higher-order differential equations: involve second, third, or higher-order derivatives, e.g., $\frac{d^2y}{dx^2} + y = 0$.

3. By linearity:

   • Linear differential equations: can be written as a linear combination of the unknown function and its derivatives, e.g., $$a_0(x)y + a_1(x)\frac{dy}{dx} + \dots + a_n(x)\frac{d^n y}{dx^n} = g(x).$$
   • Nonlinear differential equations: all others—i.e., equations where the unknown function or its derivatives appear nonlinearly (e.g., multiplied together, raised to powers, inside transcendental functions).

Real-World Example of a Differential Equation

Let’s illustrate the practical use of differential equations with a simple physics example.

While reading “An Introduction to Differential Equations: Core Concepts”, treat the accompanying diagrams as roadmap cards: first grasp the overall logical flow, then examine why each step follows, and finally verify whether boundary or initial conditions are properly accounted for.

Example: Simple Radioactive Decay

Radioactive decay of a substance is modeled by a first-order linear differential equation. Let $N(t)$ denote the amount of substance present at time $t$. According to the radioactive decay law:

$$\frac{dN}{dt} = -kN,$$

where $k > 0$ is the decay constant.

Solving this differential equation yields an explicit expression for $N(t)$.

Code Example (Python)

Below is a Python implementation using the scipy library to solve the above differential equation numerically:

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# Decay constant
k = 0.5

# Differential equation model
def model(N, t):
    dNdt = -k * N
    return dNdt

# Initial condition
N0 = 10  # Initial quantity
t = np.linspace(0, 10, 100)  # Time interval

# Solve the ODE
N = odeint(model, N0, t)

# Plot the solution
plt.plot(t, N)
plt.xlabel('Time (t)')
plt.ylabel('Quantity (N)')
plt.title('Radioactive Decay')
plt.grid()
plt.show()

This code uses odeint to numerically integrate the decay model and plots how the quantity $N$ evolves over time.

If you haven’t yet fully internalized “An Introduction to Differential Equations: Core Concepts”, revisit this card and walk through its four key actions step-by-step.

When reviewing “An Introduction to Differential Equations: Core Concepts”, avoid tackling large projects upfront. Instead, start with a single, simple example to confirm whether the core logic is clear.

Summary

In this section, we introduced the fundamental concepts of differential equations—including their definition, major classifications, and a concrete application. Differential equations serve as indispensable tools for modeling dynamic systems and form the mathematical backbone for analyzing countless scientific and engineering problems.

In the next article, we will explore common techniques for solving differential equations—both analytically and numerically—to equip you with practical methods for tackling real-world problems.