Early undergraduateAnalysisPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
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, 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
Optional discoveries
- Stochastic processes Adding random noise to differential equations leads to stochastic differential equations.
Learning pathways
Further reading
- Morris W. Hirsch, Stephen Smale and Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos (2004)
- Steven H. Strogatz, Nonlinear Dynamics and Chaos (1994)