CHAPTER 06

Differential Equations

Use derivatives to describe how quantities change and how mathematical systems evolve over time.

\[ \text{quantity} \rightarrow \text{rate of change} \rightarrow \text{differential equation} \rightarrow \text{solution} \]

DIFFERENTIAL EQUATIONS

Equations that describe change

Understand ODEs as equations that describe an unknown function through its rates of change.

Solve suitable first-order equations by separating dependent and independent variables.

Use starting values to select a particular solution from a family of ODE solutions.

Solve equations of the form \(y'+P(x)y=Q(x)\) using an integrating factor.

Study equations whose rate of change depends on the current state rather than explicitly on time.

Find equilibria and determine whether nearby solutions move toward or away from them.

Study equations involving second derivatives, characteristic equations and oscillatory solutions.

Describe several interacting quantities whose rates of change depend on one another.

Approximate solutions using Euler's method and understand the idea behind more accurate methods.

Translate assumptions about rates, flows and interactions into deterministic mathematical models.

\[ \frac{dy}{dt}=f(t,y) \]

A differential equation tells us how a quantity changes. Solving it means finding the function whose evolution follows that rate-of-change rule.