1: First order ODEs
- Page ID
- 330
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Integrals as solutions
- This document provides an overview of solving first-order ordinary differential equations (ODEs), particularly when the function \(f\) depends only on the variable \(x\). The process involves integrating \(f(x)\) to find the general solution, represented as \(y(x) = \int f(x) dx + C\). It discusses the relationship between definite and indefinite integrals and includes methods to solve equations of the form \(y' = f(y)\).
- 1.2: 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.
- 1.3: Separable Equations
- The page discusses solving different types of differential equations, specifically focusing on integrating to find solutions for separable equations. It explains that when equations are separable, i.e., \(y' = f(x)g(y)\), both sides can be integrated separately to potentially solve for \(y\). Examples are provided to illustrate solving differential equations using integration, implicit solutions, and the challenges in finding explicit solutions.
- 1.4: 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.
- 1.5: Substitution
- This page discusses the method of substitution in solving differential equations, illustrating how changing variables can simplify complex equations. It explains substitution techniques for various forms, such as Bernoulli and homogeneous equations, and provides examples to demonstrate when and how to apply these substitutions effectively.
- 1.6: Autonomous equations
- The page discusses autonomous differential equations, emphasizing their independence from time. It explains Newton's law of cooling and the logistic equation, highlighting equilibrium solutions and critical points. The stability of these points is explored through the behavior of solutions over time. A phase diagram is introduced as a tool to visualize solution behavior without solving equations exactly.
- 1.7: Numerical methods: Euler’s method
- The text discusses the challenges of solving differential equations that cannot be solved in closed form. It introduces Euler's Method as a basic numerical approximation method and explains its application in solving \(y' = \frac{y^2}{3}\). The text outlines the computation steps using different step sizes \(h\) and explains the error behavior, including how halving the interval generally halves the error.
- 1.8: Exact Equations
- This page discusses the concept of exact equations in physics and engineering, focusing on the identification and usage of potential functions related to conservation laws, such as the conservation of energy. It provides a detailed explanation of how to derive exact differential equations from potential functions, solve them, and interpret these solutions in terms of path independence and conservative vector fields.
- 1.9: First Order Linear PDE
- The page delves into solving linear first-order partial differential equations (PDEs), focusing on the transport equation where \(u_t + \alpha u_x = 0\). It discusses the use of characteristic curves to transform PDEs into ordinary differential equations (ODEs), simplifying them for solutions.
- 1.E: First order ODEs (Exercises)
- These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.