The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves
In this lesson, the geometrical view of differential equations is explored, specifically direction fields and integral curves. The lecture begins by assuming the viewer already knows how to separate variables and solve basic physical problems with differential equations. The focus is on first-order ODEs, which are written with the derivative of y isolated on the left and everything else on the right. The lecture explains how to draw a direction field and integral curves, and how they relate to finding solutions to differential equations. The importance of understanding the geometrical view of differential equations is emphasized, especially for blue equations that cannot be solved analytically.
The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves -- Lecture 1. Understanding the geometrical view of differential equations.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 23, 2008). License: Creative Commons BY-NC-SA.
More info at: http://ocw.mit.edu/terms
- Notes - Notes and exercises written by Prof. Mattuck
- Recitation Problems - Recitation Problems and classwork exercises
- Recitation Solutions - Recitation problem solutions
- Separable Differential Equations - A tutorial on separable differential equations.
- Lecture Notes - Lecture Notes
- Supplementary Notes - Chapter 1 Supplementary Notes written by Prof. Miller
- What are differential equations?
- What are first-order differential equations?
- What are ordinary differential equations?
- What are differential equations?
- What are first-order differential equations?
- What are ordinary differential equations?
- What are ODE's?
- What are separable differential equations?
- What differential equations are not solvable?
- What are direction fields?
- What are integral curves?
- How do you draw a direction field?
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Staff Review
This is a really great video introduction to ordinary differential equations. It explains exactly the geometric interpretation of what is going on. This lesson gives you a very complete, over-arching view of what differential equations look like in a plane. Also, direction fields and integral curves are discussed and drawn for equations.
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