Wright Math

Okay, the task before us now is, given a differential equation, say, dy/dx = x + y, is to visualize the solution curves (so, y = f(x)), without having to actually solve the DE.

Why is this important? 

It turns out that many DE's can't be solved analytically (meaning, with a defined algebraic form), but any first order DE can graphed as a "Slope Field" (which our textbook calls a "Direction Field").  

Another important reason (and why I wanted folks to see this material right before moving on to Multivariable/Vector Calculus) is that this concept gets extended to Vector Fields which are hugely important in physics, biology, medicine, Earth Science...on and on.  Vector Fields look just like Slope Fields, except Vector Fields take into account magnitude and direction of all the little slope lines.

This link will take you to an example from Neuroscience where you'll see images showing vector field modeling of flow of activity in the brain.  Pretty cool!  

This next series of videos will look at just Slope Fields with just a suggestion about what the Solution Curves might look like.

Videos:  

Slope (Direction) Fields Intro- Note:  Sal is using x and y.  Our book uses t and y, which I'm not a fan of.  You can substitute x for t in the homework problems if you like.

Slope at a Single Point (detail)

Matching Field and DE Example

Slope (Direction) Fields Example

Practice

The following videos will examine how to sketch Solution Curves in a Slope Field.

Note:  Just to make sure you've got the vocabulary down:

DE = Differential Equation

IC= Initial Condition (a specific point on the solution curve)

IVP = Inital Value Problem = DE + IC

Videos:  

Sketching Solution Curves

Solution Curve for IVP

Practice

Once you have the basic idea down about how the graph of a Slope Field is generated, you'll want to use technology to do this efficiently.  Here is a Slope Field grapher that will allow you to put in initial conditions (IC) to see what the solution curves would look like for various IC's.

Slope Field Generator with Solution Curves

Assigned Problems