स्नातक (प्रारंभिक)Analysisपाठ उपलब्ध
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
इस पर आधारित
36 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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).
आम भ्रांतियाँ
- 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.
आगे ले जाता है
वैकल्पिक खोजें
- Stochastic processes Adding random noise to differential equations leads to stochastic differential equations.
अधिगम पथ
आगे पढ़ने के लिए
- 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)