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Chapter 1: Differential Equations - Introduction ¶

The Language of Change¶

Differential equations describe how things change in the world.

What is a Differential Equation?¶

An equation involving:

A function (e.g., temperature T(x,t))
Its derivatives (e.g., dT/dt, d²T/dx²)

Example: Newton’s Second Law $$ F = ma = m\frac{d^2x}{dt^2} $$

This relates force to the second derivative of position!

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from matplotlib.animation import FuncAnimation
from IPython.display import HTML

sns.set_style('whitegrid')
plt.rcParams['figure.figsize'] = (14, 10)
np.set_printoptions(precision=3, suppress=True)

Types of Differential Equations¶

Ordinary Differential Equations (ODEs)¶

One independent variable (usually time t)

Example: Exponential growth/decay $$ \frac{dP}{dt} = kP $$

Solution: P(t) = P₀e^(kt)

Partial Differential Equations (PDEs)¶

Multiple independent variables (e.g., space x and time t)

Example: Heat equation $$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} $$

def exponential_growth():
    """Visualize exponential growth solution."""
    t = np.linspace(0, 5, 100)
    
    fig, axes = plt.subplots(1, 2, figsize=(14, 6))
    
    # Different growth rates
    ax1 = axes[0]
    for k in [0.5, 1.0, 1.5, 2.0]:
        P = np.exp(k * t)
        ax1.plot(t, P, linewidth=2, label=f'k = {k}')
    
    ax1.set_xlabel('Time t', fontsize=12)
    ax1.set_ylabel('Population P', fontsize=12)
    ax1.set_title('Exponential Growth: dP/dt = kP', fontweight='bold')
    ax1.legend()
    ax1.grid(True, alpha=0.3)
    
    # Decay
    ax2 = axes[1]
    for k in [-0.5, -1.0, -1.5]:
        P = np.exp(k * t)
        ax2.plot(t, P, linewidth=2, label=f'k = {k}')
    
    ax2.set_xlabel('Time t', fontsize=12)
    ax2.set_ylabel('Population P', fontsize=12)
    ax2.set_title('Exponential Decay', fontweight='bold')
    ax2.legend()
    ax2.grid(True, alpha=0.3)
    
    plt.tight_layout()
    plt.show()
    
    print("Solution: P(t) = P₀ e^(kt)")
    print("  k > 0: Growth")
    print("  k < 0: Decay")

exponential_growth()

Why Differential Equations Matter¶

Natural Phenomena¶

Physics: Motion, heat, waves, quantum mechanics
Biology: Population dynamics, disease spread
Chemistry: Reaction rates
Engineering: Circuits, control systems

The Power¶

Once you write down the differential equation:

It captures the rule of how things change
Solving it predicts the future
It works for any initial condition!

Next: The heat equation - a beautiful PDE!

Chapter 1: Differential Equations - Introduction¶