Overview
Given a differential equation \(\displaystyle \frac{dy}{dt} = f(t,y)\), we can evaluate \(f\) at a point \((t,y)\) to find the slope of the tangent (the derivative) of a solution \(y=y(t)\) which passes through this point. Finding the slopes of the tangent line for a wide selection of points in the plane give us a so-called slope field. With this slope field in hand, we can describe the general behavior of solutions of the differential equation without actually explicitly solving the differential equation. (An approximative method to solve a differential equation based on this approach is called “Euler’s Method”.)
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
-
Use a differential equation to calculate slopes of solutions at given points, and describe these slopes in terms of the physical situation modeled by the differential equation.
-
Explain the construction of a slope field. Specifically, explain how slope fields are related to tangent lines, and how a differential equation can be used to create a slope field.
-
Graphically sketch a solution to a differential equation when given all or part of a slope field.
-
State the meaning of equilibrium solution, and find them for simple differential equations.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
-
Construct a slope field by hand.
-
Give qualitative descriptions of solutions to differential equation using data given by a slope field.
-
Calculate and interpret isoclines: The curves for which a differential equation has a given slope.
-
Find and describe stable and unstable equilibria.
To prepare for class
-
Do the Preview Activity for section 7.2 (on WeBWorK if required by your teacher).
-
Watch the following videos (by GVSUmath):
-
Do some experimentation with the following interactive applet which allows you to enter a formula \(f(x,y)\), and then draws the corresponding slope field \(\frac{dy}{dx}=f(x,y)\) as well as some approximative solutions (through points which you can move around):
Slope Field Plotter (by Dr. Adrian Jannetta)
After class
-
Do the Preview Activity for section 7.3 (on WeBWorK if required by your teacher).
-
Do some more experimentation with the interactive applet listed above, realize that it actually uses Euler’s method to find the approximative solutions, and play around with changing the “step size”.
