Ordinary Differential Equations

Master the theory and techniques of ordinary differential equations, from elementary methods to advanced stability analysis and applications in physics and engineering.

Course Overview

Prerequisites

Calculus (differentiation and integration), basic linear algebra (matrices, eigenvalues)

What You'll Learn

Solution techniques, existence theorems, stability analysis, and real-world applications

Duration

9 courses covering essential ODE theory and applications

Course Modules

ODE-01

Introduction and Elementary Methods

ODE basics, direction fields, separable equations, homogeneous equations, and population models.

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ODE-02

First-Order Linear Equations

Linear equations, integrating factor method, Bernoulli equations, Riccati equations, and applications.

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ODE-03

Exact Equations and Integrating Factors

Exact equations, integrating factors, implicit solutions, and Clairaut equations.

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ODE-04

Higher-Order Linear Equations

Reduction of order, first integrals, autonomous equations, and phase plane analysis.

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ODE-05

Constant Coefficient Linear Equations

Characteristic equations, undetermined coefficients, variation of parameters, and operator methods.

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ODE-06

Linear Systems of ODEs

Matrix exponential, fundamental matrices, Jordan form, and variation of parameters for systems.

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ODE-07

Existence and Uniqueness Theory

Picard theorem, Lipschitz condition, Gronwall inequality, and solution extension theorems.

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ODE-08

Stability and Lyapunov Methods

Stability definitions, linear approximation, Lyapunov functions, and asymptotic stability.

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ODE-09

Special Equations and Applications

Bessel equations, Sturm-Liouville problems, epidemiological models, and physical applications.

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Recommended Learning Path

Foundation (ODE-01 to ODE-03)

Master first-order equations, elementary methods, exact equations, and integrating factors.

Higher-Order Theory (ODE-04 to ODE-06)

Learn higher-order equations, constant coefficient methods, and linear systems.

Advanced Theory (ODE-07 to ODE-09)

Explore existence theory, stability analysis, and special equations with applications.

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