Early undergraduateAnalysisLesson available

Ordinary differential equations

First-order equations, linear equations with constant coefficients, linear systems and phase portraits via eigenvalues, existence and uniqueness theorems, and qualitative behaviour.

math.analysis.ordinary-differential-equations

Open lesson Show everything needed to reach this topic

Builds on

Calculus of one variable Linear algebra

Required prerequisites, all the way down: 36

You will learn to

Solve separable equations, linear first-order equations and constant-coefficient linear equations.
Solve linear systems x′ = Ax using eigenvalues and sketch their phase portraits.
State the Picard–Lindelöf theorem and give an example of non-uniqueness when its hypotheses fail.

Definition

An ordinary differential equation of order n relates an unknown function of one variable to its first n derivatives.
An equilibrium of x′ = f(x) is a point where f vanishes.

Theorem

Picard–Lindelöf: if f is continuous and locally Lipschitz in x (uniformly in t), then x′ = f(t, x), x(t₀) = x₀ has a unique solution on some interval around t₀.
The solution of x′ = Ax with initial value x(0) is x(t) = e^(tA) x(0).
x′ = √|x| with x(0) = 0 has more than one solution, e.g. x ≡ 0 and x(t) = t²/4 for t ≥ 0 (and 0 for t < 0).

Common misconceptions

Assuming solutions exist for all time; solutions of x′ = x² can blow up in finite time.
Thinking every differential equation has a solution in closed form.

Leads to

Classical mechanics Partial differential equations and Fourier series Numerical analysis Mathematical biology Dynamical systems and chaos

Optional discoveries

Stochastic processes Adding random noise to differential equations leads to stochastic differential equations.