1: First Order ODEs
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- 1.1: Introduction to Differential Equations
- This page emphasizes the importance of differential equations in science and engineering, focusing on their role in modeling physical laws and solving first-order equations. It includes the derivation of a bacteria population growth model, showcasing calculations for growth rates and populations over time. The text also discusses solution types for differential equations, including general and particular solutions, while connecting hyperbolic functions to real-world shapes.
- 1.2: Classification of Differential Equations
- This page discusses the classification of differential equations as ordinary or partial based on independent variables, emphasizing the importance of order and linearity. It covers concepts of homogeneous vs. nonhomogeneous equations, constant vs. variable coefficients, and introduces autonomous equations. Additionally, it mentions a 7-minute video providing context for understanding these concepts and solutions.
- 1.3: Integrals as Solutions
- This page discusses solving first-order ordinary differential equations (ODEs) using integration, detailing the standard form and methods for general solutions. It differentiates between indefinite and definite integrals, supplemented by examples illustrating ODE applications in mechanics. Additionally, it provides two calculus problems involving a car's speed and acceleration, showcasing how to calculate distance through integration over specified time intervals.
- 1.4: Slope Fields
- The general first order equation we are studying looks like y′=f(x,y). In general, we cannot simply solve these kinds of equations explicitly. It would be nice if we could at least figure out the shape and behavior of the solutions, or if we could find approximate solutions.
- 1.5: Separable Equations
- This page discusses separable differential equations, emphasizing identification, variable separation, and the integration process for solutions. It addresses implicit solutions, their importance, challenges in deriving explicit forms, and includes examples demonstrating the method.
- 1.6: Linear Equations and the Integrating Factor
- One of the most important types of equations we will learn how to solve are the so-called linear equations. In fact, the majority of the course is about linear equations. In this lecture we focus on the first order linear equation.
- 1.7: Existence and Uniqueness of Solutions of Nonlinear Equations
- Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximations, it is useful to know conditions that imply the existence and uniqueness of solutions of initial value problems for nonlinear equations. In this section we state such a condition and illustrate it with examples.
- 1.8: Substitution
- This page covers methods for simplifying non-separable differential equations through substitution, particularly Bernoulli equations, highlighting the significance of making effective substitutions to achieve linear forms for easier solutions. It provides examples demonstrating the use of the substitution v=yx for homogeneous equations, illustrating the process of transforming complex equations into simpler, separable ones.
- 1.9: Autonomous Equations
- This page analyzes autonomous differential equations focusing on dependent variable dynamics, using examples like Newton's law of cooling and the logistic equation to discuss critical points and their stability. It emphasizes the importance of critical points in population dynamics, illustrating how parameter variations affect outcomes, including extinction or stabilization.
- 1.10: Numerical Methods - Euler’s Method
- This page elaborates on Euler's method for approximating solutions to differential equations when closed-form solutions are not feasible. It discusses the method's iterative approach and its first-order accuracy, noting the error reduction with smaller step sizes. The text also addresses challenges like numerical instability and the importance of selecting appropriate methods and step sizes.
- 1.11: Exact Equations
- This page discusses exact differential equations, their solutions, and the concept of potential functions in physics, emphasizing the total derivative's role. It illustrates the Poincaré Lemma, which connects local potential functions to exact equations, and addresses the use of integrating factors to solve non-exact equations.
- 1.12: Transformation of Nonlinear Equations into Separable Equations
- This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.