Chapter 8 Lesson 33: Analytical Solutions II

8.1 Objectives

  1. Given an IVP, use separation of variables to find the general and specific solutions by hand.

  2. Given an IVP in the context of a real‐world scenario, find the solution and interpret its results within the context of the scenario.

8.3 In class

  1. Review. This is a continuation of last lesson. Review separability, and how we can not find a more simple expression for \(\int y(t)\,dt\), at least without knowing what \(y(t)\) is, but we *can find a simpler expression for \(\int y(t)\frac{dy}{dt}\,dt\).

  2. Practice. Much of this section should be devoted to practicing finding solutions to ODEs of the form \(y^{'} = f(y)\). There is time for you to introduce your own examples, or you can work with the ones from the book.

  3. Applications Spend a significant amount of time on the application problems in the back of the book. Students should be able to

    1. Identify the correct initial conditions from a word problem.
    2. Use the solution to the IVP to answer a context-specific question.
    3. Students are welcome to solve these questions using either algebra, or by using tools like findZeros. Both are demonstrated in the book.

8.4 R Commands

antiD, findZeros

8.5 Problems & Activities

  1. Spend about half the class having the students solve exercises from the back of the chapter. They can use antiD if they need to.

  2. Cover the two “application” problems from the book, 7.4.3 and 7.4.4, paying special attention to the “Additional Insights”. Students will need to be able to identify the correct initial conditions from a word problem, solve the IVP, and then use the solution to the IVP to solve an additional question. Try to cover the full examples, as they will be asked similar questions in the homework. More examples can be found in HW 35, or swing by my office and borrow an ODE book.