Mathematics & Science
Learning Center |
|||
Slope Fields with MathematicaMaking Slope Fields by YourselfA Post Exercise Discussion ofdy/dx = sin x, on the region -2π ≤ x ≤ 2π, and -5 ≤ y ≤ 5Once again, I hope you actually did do the exercise in Mathematica before you came here to see the discussion. If you didn't do it yet , go back and do the exercise now! Assuming that you did the exercise correctly, you should have produced a picture that was very much like the following: Here too, you should have noticed that the isoclines for this slope field are vertical lines, (just like the last example.) Do you recognize the general trend of the field this time? It's harder to see than the one with the parabolas, but if you really think about it, those little waves should look a bit familiar. Hmmm, symmetry about the y-axis, like the cosine function, but upside down. That would make it the negative of the cosine function, wouldn't it? A whole family of them shifted vertically from one another! Again, this makes sense if you think about solving this differential equation using integration. If you simply integrated: You'd get: This would make a whole family of upside down cosine functions, depending upon what value you give to the constant C, which would match what the slope field is telling us too. Are you starting to make the connection between slope fields and analytic solutions? That's the end of my discussion of this problem, so let's go back to the exercises. |
|||
ODE Laboratories: A Sabbatical Project by Christopher A. Barker ©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |