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所在平台: Udemy |
课程主页: https://www.udemy.com/course/intro-to-differential-equation/
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Coursera 上的《微分方程导论》是一门为初学者设计的入门课程。课程内容涵盖了微分方程的基础知识，将其分为六个主要部分： 1. **微分方程导论：** 课程首先定义了微分方程（包含一个或多个因变量相对于一个或多个自变量的导数的方程），并介绍了其基本术语和分类，包括区分线性与非线性方程、常微分方程 (ODE) 与偏微分方程 (PDE)，以及根据阶数和线性度对 ODE 进行分类。此外，课程还讲解了初值问题，并区分了一阶和二阶微分方程。 2. **一阶微分方程：** 这一部分将深入探讨一阶微分方程的各种解法和概念，包括方向场、可分离方程、欧拉法、一阶线性微分方程的求解、精确 ODE 的求解、齐次微分方程、伯努利微分方程，以及线性与非线性模型（如捕食者-猎物模型和电路网络）的应用。 3. **二阶微分方程：** 课程将介绍二阶微分方程的求解技术，重点关注不同类型的根（实根、复根、重根），并引入朗斯基行列式 (Wronskian)。还将讲解待定系数法、微分算子、参数变分法以及柯西-欧拉方程。机械振动模型的应用也将被讨论。 4. **级数解：** 本部分将侧重于使用幂级数方法求解微分方程。 5. **拉普拉斯变换：** 课程将详细介绍拉普拉斯变换及其应用，包括拉普拉斯变换的概述、逆拉普拉斯变换、利用拉普拉斯变换解决初值问题，以及拉普拉斯变换的平移定理。 6. **微分方程组：** 最后，课程将涉及微分方程组，包括齐次线性系统和相平面图的分析。
This is a lower-level introduction to differential equation course. This course is broken into 6 sections: 1) Intro to differential equations, 2) First order differential equations, 3) Second order differential equations, 4) Series solution, 5) Laplace transform. First, we will go over definitions and terminology, we will go over the difference between linear non linear equations. Then storm into initial value problems, go over the difference between first order DE, second order DE, ordinary differential equations, classify ODE by order and linearity.The first thing we will go over is define a differential equation. A differential equation is an equation containing the derivatives of one or more dependent variables with respect to one or more independent variables(equation involving some unknown function and one or more of its derivatives). In a differential equation the solution is always going to be an equation. There are two types of differential equations: ordinary differential equation, partial differential equation. Also a differential equation can be linear or nonlinear.First Order Differential Equations:Direction fieldsSeparable equationsEuler's methodFirst order linear differential equationSolving exact ODE'sHomogeneous Differential EquationBernoulli differential equationLinear and nonlinear modelsPredator prey models and electrical newtworksSecond Order Differential EquationsDistinct real rootsComplex rootsRepeated rootsWronskianUndetermined coefficientsDifferential operatorsVariation of parametersCauchy euler equationMechanical vibrationsSeries SolutionsPower seriesSeries SolutionsLaplace TransformLaplace transform overviewInverse laplace transformsInitial value problems with laplace transformsTranslation theorems of laplace transformsSystems Of Differential EquationsHomogeneous linear systemsPhase plane portraits
