Doing so we arrive at the following image: As before, we let Mathematica do the heavy lifting. Sadly we have a hard time visualizing this one too. This problem, in implicit form, is: x 2 sin y + y 3 = C. Using this technique, it can be shown that the actual analytic solution to Technique you'll see that exact differential equations aren't too hard to Simple algebra and simple calculus for solution. We covered in the previous problem, this method relies on a little more than just Unfortunately, unlike separation of variables, which Learned how to solve exact differential equations analytically already in the What about finding an analytic solution? This problem is of a class ofĭifferential equations referred to as exact. Don't forget to return here when you are done. Feel free to experiment with it a little. If you do, please try to read it carefully, and try to understand what each command is being used for. You will not be asked to graph isoclines like this on lab exams, but if you'd like to see the Mathematica notebook that was used to make the above plot, click on the icon to the left. Try to verify from this image that as you trace along any particular one of the red isoclines, the slopes of the vectors which intersect it are all the same. Once all of this hard work is over, the resulting graph looks something like this: To get decent looking images, it helps to have Mathematica do a separate plot for each quadrant, and then combine these together for the final image. Furthermore, the isocline curve families don't plot well, as they fall into distinct sub-families for each quadrant of the plane. This time the required work is much more complicated, since the isoclines found above are defined implicitly, and to make Mathematica plot implicitly defined curves requires the loading of yet another external package. Mathematica help us with the graphing difficulties. To get much insight from the isocline equation would require having I personally don't have a clue what the graph of this equation looks like.

Isoclines analytically, (the same way we did in the last two exercises), leads Produced a picture that was a version of the following:Īs you can see, once again neither horizontal nor vertical lines are isoclines. Slope Fields with Mathematica Making Slope Fields by Yourself A Post Exercise Discussion of on the region -3 ≤ x ≤ 3, andĭon't cheat! If you didn't do the exercise in Mathematica before youĬame here to see the discussion, go back and do itĪssuming that you did the exercise correctly, you should have

Wolfram mathematica plot vertical line