Lecture Notes: Introduction to Differential Equations

Key Quote

"Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations." - Steven Strogatz

Overview

• Differential equations describe change rather than absolute amounts.
• Useful in various fields, not just physics.
• Aim of lecture series: Provide a big picture view of differential equations while exploring detailed examples.

Prerequisites

• Basic calculus knowledge (derivatives and integrals).
• Basic linear algebra will be required later, but not extensively.

What are Differential Equations?

• Appropriate when describing change is easier than stating absolute values.
• Examples include:
  • Population growth or decline.
  • Changes in personal affection.

Types of Differential Equations

1. Ordinary Differential Equations (ODEs)
   • Functions with a single input (often time).
   • Example: Motion described in terms of force leading to acceleration.
2. Partial Differential Equations (PDEs)
   • Functions with multiple inputs.
   • More complex, often involving continuous values changing over time.
   • Will be discussed in later videos.

Example: Trajectory of a Thrown Object

• Force of gravity causes downward acceleration of 9.8 m/s² (denoted as g).
• Differential equation for vertical position (y):
  • y'' = -g
• Solving the equation:
  1. First derivative (velocity): y' = -gt + v₀
  2. Second derivative (position): y = -1/2 gt² + v₀t + y₀

Motion Under Variable Forces

• For planetary motion, gravity is not constant (depends on distance between bodies).
• Motion involves interaction between two variables (position and velocity).
• Typically involves second-order differential equations.

Example: Pendulum Dynamics

• The angle (θ) changes as a function of time, representing harmonic motion.
• Pendulum period approximation: T = 2π√(l/g)
  • Where l = length of the pendulum.
• Actual behavior deviates from simple sine wave predictions for larger angles.
• Differential equation setup:
  • x = Lθ (position along arc)
  • Acceleration: x'' = -g sin(θ)
  • Including air resistance: x'' = -g sin(θ) - μx'

Challenges of Solving Differential Equations

• Complex solutions (especially with damping terms) often can’t be expressed in simple terms.
• Visualization through phase space: a state is represented by angle and angular velocity.
• Phase diagrams can help understand system behavior over time.

Stability and Fixed Points

• Stability questions can arise, asking if nudges lead to a return to an equilibrium state.
• Fixed points correspond to stable (pendulum at rest) and unstable (pendulum upright) positions.

Love as a Mathematical Model

• Relationship dynamics can be modeled similarly to physical systems:
  • Affection changes based on partner’s feelings.
  • Similar phase space structures emerge in both models.

Numerical Solutions

• Simulating the system using finite time steps provides an approximation of solutions.
• Basic numerical method: incrementally update position and velocity based on the differential equation.
• More sophisticated methods exist for balancing accuracy and efficiency.

Chaos Theory

• Some systems exhibit sensitivity to initial conditions (e.g., three-body problem).
• Small variations can lead to vastly different outcomes, posing challenges for prediction.

Conclusion

• Differential equations form a mathematical framework for understanding change.
• Despite their challenges, they offer a pathway to explore the complexity of dynamic systems in the world.