In Unit I, we will study ordinary differential equations (ODE’s) involving only
the first derivative.
\(y’ = F(x, y)\)
The first session covers some of the conventions and prerequisites for the
course. After that we will focus on first order differential equations. We’ll
start by defining differential equations and seeing a few well known ones from
science and engineering. Very quickly we will learn about the three main ways of
approaching ODE’s: Analytic (in symbols), geometric (with pictures and graphs),
and numerical (with the computer).
A large part of this unit (and this course) will be spent on linear ODE’s. These
are fundamental to the understanding of all ODE’s and among the most tractable.
Last Modification: 2020-09-16 Wed 19:55
- Captured On
- [2019-09-17 Tue 16:00]
- Source
- Unit I: First Order Differential Equations | Differential
Equations | Mathematics | MIT OpenCourseWare
Captured On [2019-09-17 Tue 16:00] Source Basic DE’s and Separable Equations | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 Can be \(e^{at}=0\)? 1.1 Front Can be \(e^{at}=0\)?
1.2 Back Never
2 Write a sketch of the graph 2.1 Front Write a sketch of the graph
\(y=e^t\)
2.2 Back 3 Write a sketch of the graph 3.1 Front Write a sketch of the graph...
Captured On [2019-09-17 Tue 16:00] Source Geometric Methods | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a direction field 1.1 Front What is a direction field
for the equation \(y’ = f(x,y)\)
1.2 Back For each point \((x,y)\) of the plane is drawn a little segment whose slope is \(f(x,y)\)
For example: \({\displaystyle \dv{y}{x} = 2x}\)
2 How can we draw a direction field by hand?...
Captured On [2019-09-11 Wed 20:32] Source First Order Linear ODE’s | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a first order linear ODE? 1.1 Front What is a first order linear ODE?
Write the standard form
1.2 Back In the unknown function \(x = x(t)\)
\({\displaystyle A(t) \dv{x}{t} + B(t) x(t) = C(t)}\)
As \(A(t) \neq 0\), we can simplify the equation by dividing by \(A(t)\)...
Captured On [2019-09-17 Tue 15:58] Source Complex Arithmetic and Exponentials | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a complex number? 1.1 Front What is a complex number?
1.2 Back A complex number is an expression of the form \(a + ib\)
2 When can we say that 2 complex number are equals? 2.1 Front When can we say that 2 complex number are equals?...
Captured On [2019-09-26 Thu 15:47] Source Sinusoidal Functions | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a sinusoidal function equation? 1.1 Front What is a sinusoidal function equation?
1.2 Back \(f(t) = A \cos(\omega t - \phi)\)
2 What is the sinusoidal oscillation equation? 2.1 Front What is the sinusoidal oscillation equation?
2.2 Back \(f(t) = A \cos(\omega t - \phi)\)...
Captured On [2019-09-27 Fri 19:04] Source First Order Constant Coefficient Linear ODE’s | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a constant coefficient First ODE? 1.1 Front What is a constant coefficient First ODE?
1.2 Back \(\dot{y} + ky = q(t)\), where \(k\) is a constant
2 What is the solution of a constant coefficient First ODE? 2.1 Front What is the solution of a constant coefficient First ODE?...
Captured On [2019-10-01 Tue 12:22] Source Exponential Input; Gain and Phase Lag | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 Explain the method of optimism to solve this first order ODE 1.1 Front Explain the method of optimism to solve this first order ODE
\(\dot{x} + 2x = 4e^{3t}\)
1.2 Back The inspiration is based on the fact that \(\dv{t} e^{rt} = re^{rt}\)....
Captured On [2019-10-02 Wed 13:06] Source First Order Autonomous Differential Equations | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is an autonomous first order differential equation? 1.1 Front What is an autonomous first order differential equation?
1.2 Back There are (in general) nonlinear equations of the form \(\dot{x} = f(x)\)
The word autonomous means self governing and indicates that the rate of change of \(x\) is governed by \(x\) itself and it not dependent of time....
Captured On [2019-10-07 Mon 13:29] Source Linear vs. Nonlinear | Unit I: First Order Differential Equations | Differential Equations | Mathematics | MIT OpenCourseWare