本科低年级Analysis计划中

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

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前置知识

Calculus of one variable Linear algebra

36 项必需的前置知识（一直追溯到底）

你将学会

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.

定义

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.

定理

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).

常见误解

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.

通向

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

可选发现

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

学习路径

Applied mathematics