Class Summary Page

for

MATH 362

Differential Equations

  • Monday, January 27. We went over the syllabus and discussed the assumptions that went into the Exponential Growth Law (Malthus) and the Logistic Growth Law (Verhulst). We talked about the difference between ordinary and partial differential equations; in particular, we discussed how to compute partial derivatives. A self-diagnostic tool for review of material from MATH 160 and Math 162 was distributed.

  • Wednesday, January 29. We defined the order of an ODE, and discussed the difference between the general and simplified forms. We worked a word problem involving a simple second-order differential equation (similar to a homework problem on drag racing). This example indicated to us that (i) we should have to do n integrations to solve a nth order differential equation, (ii) we will need n extra conditions to get from the general solution of the differential equation to the special solution (of the initial value problem) that is of interest to us, and (iii) in real-world problems, obtaining this special solution might only be one part of the process of answering whatever the real question is.

  • Friday, January 31. We began by talking some more about applications involving second order equations of the form y''=f(x): drag racing and the quiz problem. We worked a word problem involving a simple second-order ODE (similar to a homework problem on drag racing). We worked through some examples that led us to our definition of an explicit solution to a differential equation: a function that satisfies the equation and an open interval on which that function satisfies the equation. We discussed the antiderivative notation that we will use in the course. Finally we introduced the notion of a separable ODE, and tested several equations for separability.

  • Monday, February 3. We worked through a Maple notebook, "Verifying Explicit and Implicit Solutions to ODEs." We learned to be wary of what Maple tells us. For instance, in one problem when we asked Maple to solve an implicit relation, it gave us an answer in terms of functions that we didn't know. When we later graphed the solution curves, we saw that it had neglected part of the solution (without telling us that it did this). At another point, one of the plots was entirely incomprehensible.

  • Wednesday, February 5. We solved the Exponential Growth Law, and began solving the Logistic Growth Law. Our solution method for these separable equations requires us to be be able to employ various integration techniques. For instance, today's class ended with a quick review of partial fractions.

  • Friday, February 7. We obtained the complete solution to the Logistic Growth Equations, and began to study graphical methods for understanding the behavior of solutions to first-order equations.

  • Monday, February 10. We worked more on graphical methods. We saw how to make this a systematic procedure by first looking for isoclines, and then using these to sketch the direction field.

  • Wednesday, February 12. We used Maple --- we worked our way through the notebook "Direction Fields" --- to help us sketch the direction fields and solution curves of first-order ODEs.

  • Friday, February 14. We discussed the Existence and Uniqueness Theorems for first-order ODEs. An example from physics (water draining froim a tank --- Torricelli's Law) was used to show how and why one might encounter the situation of nonuniqueness in the real world.

  • Monday, February 17. A handout was distributed, "The Essentials of Linear Mathematics, Part I." We began discussing linear spaces by first looking at vectors in the plane, and then deciding which of their properties we should take as being the axioms of an abstract linear space.

  • Wednesday, February 19. We spent today very carefully working with various function spaces, e.g., { f | f : D -- > R } where D is nonempty, and showing that they were linear spaces. We also began to talk about subspaces, which were in another handout, , "The Essentials of Linear Mathematics, Part II."

  • Friday, February 21. We proved the Subspace Criterion which will be a very hany tool for us to use in verifying that various sets of functions are actually linear spaces. Since we can regard any set of (real-valued) functions as being a subset of { f | f : D -- > R }, in order to see that this set is actually a linear space, we only need show that it is non-empty (usually we do this by showing that it contains the zero function), that it is closed under vector addition, and that it is closed under scalar multiplication.

  • For the first half of the class, we worked through several examples using the Subspace Criterion to decide whether various sets of functions were linear spaces. In the second half of the class, we returned to the Maple notebooks, "Verifying Explicit and Implicit Solutions to ODEs" and "Direction Fields."

  • Wednesday, February 26. Today was devoted to a review.

  • Friday, February 28. The first exam was given.

  • Monday, March 3. The first exam (and a solution key) was returned. We began to discuss the notion of operators on linear spaces. We will be especially interested in studying linear differential operators. A set of examples was given, and we classified them as differential or nondifferential (easy!), and also as linear or nonlinear (harder).

  • Wednesday, March 5. A handout, "Linear Operators," was distributed. We related linear differential quations to linear differential operators. We saw that any nth order linear differential equation could be written more compactly as L[y]=F for some appropriate nth order linear differential operator L. We ended with the observation that for any linear operator L[0]=0, so if we ever are given an operator T for which T[0] is not 0, then we know that T is nonlinear.

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