First-Order Ordinary Differential Equations

First-Order Ordinary Differential Equations

General Form

Existence and Uniqueness

Separable Equations

Solution:

Linear First-Order Equations

Solution using integrating factor:

Exact Equations

Solution: Find with , then .

Bernoulli Equations

Substitution reduces to a linear equation.

Homogeneous Equations

Substitution: , so and .

Examples

Example 1: (separable)

Example 2: (linear)

Integrating factor:

Applications

Direction Fields

Autonomous Equations

Phase line analysis: equilibrium solutions are where . Classification as stable, unstable, or semi-stable depends on .