Ordinary differential equations

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 36

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• 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).

مفاهيم خاطئة شائعة

• 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.

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اكتشافات اختيارية

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

مسارات التعلّم

• Applied mathematics

قراءات إضافية

• 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)