Differential Equations

Learning Objectives

Define a differential equation.

Distinguish between ordinary differential equations (ODEs) and partial differential equations (PDEs).

Solve a simple separable first-order ODE.

Describe the application of differential equations in modeling real-world phenomena.

The Mathematics of Change

A differential equation is an equation that relates a function with its derivatives. Whereas algebraic equations have numerical solutions (e.g., x=2), differential equations have functions as solutions. They are fundamental to modeling dynamic systems in science and engineering.

Types of Differential Equations

Ordinary Differential Equation (ODE): An equation involving a function of only one independent variable and its derivatives.

Example: Newton's second law, F=ma, can be written as F = m(d²x/dt²), where position x is a function of a single variable, time t.

Partial Differential Equation (PDE): An equation involving a function of two or more independent variables and its partial derivatives.

Example: The Heat Equation, which describes how temperature changes over both time and position.

Solving a Simple Differential Equation: Separation of Variables

This method can be used for first-order ODEs that can be algebraically separated into two parts: one with only the dependent variable (y) and its differential (dy), and the other with only the independent variable (x) and its differential (dx).

Example: Solve dy/dx = 2x/y

1.Separate variables: y dy = 2x dx

2.Integrate both sides: ∫y dy = ∫2x dx

3.Perform integration: ½y² = x² + C

4.Solve for y (optional): y² = 2x² + 2C. The solution is a family of functions.

Applications

Differential equations are the mathematical language used to describe change.

Physics: Newton's laws of motion, wave propagation, heat flow, electromagnetism.

Biology: Population growth (logistic growth model), spread of diseases, nerve impulse propagation.

Chemistry: Reaction kinetics, diffusion.

Engineering: Circuit analysis, fluid dynamics, structural mechanics.

Key Terms

Differential Equation: An equation that contains one or more functions with its derivatives.
Ordinary Differential Equation (ODE): A differential equation containing one or more functions of one independent variable and the derivatives of those functions.
Partial Differential Equation (PDE): A differential equation containing unknown multivariable functions and their partial derivatives.
Separation of Variables: A method for solving some ordinary differential equations in which one moves all terms involving the dependent variable to one side of the equation and all terms involving the independent variable to the other side.