常微分方程与边值问题

This lesson introduces mathematical modeling as the process of using differential equations to describe how physical systems evolve over time. Students learn to translate physical laws, such as Newton’s Second Law, into mathematical expressions and explore how equilibrium solutions represent the long-term behavior of dynamic systems.

This lesson explores the structural taxonomy of first-order differential equations, focusing on classifying linear, autonomous, and exact equations to model physical systems. Students learn to solve these equations using integrating factors and Euler’s method, while also examining the conditions for solution existence, uniqueness, and stability.

This lesson introduces second-order linear differential equations, focusing on the principle of superposition, existence and uniqueness theorems, and solving constant-coefficient equations using characteristic roots. Students will also learn to apply these concepts to physical vibration models and utilize the Wronskian to determine the linear independence of solution sets.

This lesson explores the transition from second-order to $n$th-order linear differential equations, emphasizing that the principle of superposition and the need for $n$ linearly independent solutions remain consistent as system complexity increases. Students learn to solve these higher-order equations using characteristic equations, accounting for repeated roots, and applying the method of undetermined coefficients to find particular solutions.

This lesson introduces power series as a method for solving differential equations that lack closed-form solutions, focusing on the concept of analyticity and Taylor series representations. Students learn to transform differential equations into algebraic recurrence relations and determine the radius of convergence based on the proximity of singular points.

Integral transforms simplify complex differential equations by mapping them from the time domain into an algebraic transform domain using a specific kernel. This lesson explores the foundations of the Laplace transform, focusing on how improper integrals and convergence criteria enable us to solve initial value problems more efficiently.

This lesson explores how to transform $n$-th order linear differential equations into systems of first-order equations by defining state-space vectors. Students learn to apply matrix algebra to solve these coupled systems, which model complex physical interactions in mechanical, fluid, and electrical systems.

This lesson introduces numerical methods as a way to approximate solutions to differential equations by discretizing the Fundamental Theorem of Calculus. Students learn how to implement the Euler method and predictor-corrector approaches while exploring the critical roles of existence theorems, step-size refinement, and numerical stability.

This lesson explores the dynamics of autonomous nonlinear systems, focusing on how critical points and phase plane analysis reveal complex behaviors that differ from linear models. Students learn to evaluate system stability using Liapunov functions, linearization, and nullcline analysis to understand the local and global topography of nonlinear trajectories.

This lesson introduces two-point boundary value problems (BVPs), which require satisfying differential equations at two distinct spatial locations rather than a single initial point. Unlike initial value problems, BVPs are sensitive to boundary conditions and may result in zero, unique, or infinitely many solutions depending on the system's parameters.

This lesson explores how physical conservation laws, such as those governing vibrating strings and electrical transmission lines, are modeled using partial differential equations. It demonstrates how the method of separation of variables transforms these equations into the generalized Sturm-Liouville eigenvalue problem, providing a unified framework for analyzing spatial dynamics.