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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
Se basa en
Requisitos previos obligatorios, hasta el fondo: 36
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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).
Errores frecuentes
- 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.
Conduce a
Descubrimientos opcionales
- Stochastic processes Adding random noise to differential equations leads to stochastic differential equations.
Itinerarios de aprendizaje
Lecturas complementarias
- 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)