| Title | Summary |
|---|---|
| First-Order: Direction Fields | Paul's On-Line Notes on direction fields for first-order equations. They work several examples, but sometimes apply them to autonomous equations, which can be analyzed faster with phase-line portraits. |
| The Geometrical View of y'=f(x,y) | Professor Mattuck (MIT video) uses graphical methods to analyze the solutions
of first-order ODEs. He uses the term standard form for what we call normal form. First, he works out the direction field for a separable equation. He then makes some contour plots of an implicit solution of that separable equation (some integral curves). Next, he works out the direction field of a nonhomogeneous linear equation. Finally, he discusses the general existence and uniqueness result given in the previous chapter of our notes and gives an example in which existence fails at many points and uniqueness fails at one point. This example is differnt than the examples illustrating the breakdown of existence and of uniquness in our notes. |
| Khan Academy: Creating a Slope Field | Khan Academy video on direction fields for a first-order equation.
It calls them slope fields. It works one example by hand. The direction field is not plotted for many points. The example is separable, and a better graphical method to apply is contour plots. |
| Khan Academy: Slope Field to Visualize Solutions | Khan Academy video on direction fields for a first-order equation.
It calls them slope fields. The direction field is generated by computer for an autonomous equation and is used to visualize some solutions. The example can be analyzed faster with phase-line portraits. |
| Khan Academy: Differential Equation from Slope Field | Khan Academy video on direction fields for a first-order equation.
It calls them slope fields. Starting with the direction field generated by computer for an unknown differential equation, the video shows how to select the correct differential equation from a list of five equations by logically eliminating incorrect answers. It then uses the direction field to visualize some solutions. |