Chapter 7 Lesson 32: Analytical Solutions II

7.1 Objectives

  1. Recognize and distinguish between autonomous ODEs of the form \(y' = f(y)\) and pure‐time ODES of the form \(y' = f(t)\).

  2. Understand the definition of a separable ODE. Given a first‐order ODE, determine if it is separable.

  3. Given a separable ODE, find the general solution using separation of variables by hand.

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

7.3 In class

  1. Separable differential equations. Up until now, we’ve only worked with ODEs whose rate of change was only dependent upon the variable \(t\), which we can solve by integration techniques from Block 4 or using antiD. The next two days will be spent working with separable ODEs. We define a first-order ODE as separable if it can be written in the form \[y'=f(t)g(y).\]

  2. Analytical solutions. In order to solve an ODE of the form \(y^{'} = f(t)g(y)\) analytically, we must use algebra to rewrite the equation. We need to get all the terms involving \(y\) multiplied by \(\frac{dy}{dt}\), and all of the \(t\) terms on the other side of the equation, not multiplied by \(\frac{dy}{dt}\). A common error cadets will make is that they will use algebra to create a sum of functions of \(y\) on the LHS which does not work. For example, consider the ODE \(y^{'} = y + 1\). In order to set up the problem correctly for integration using separation of variables so that you have a product of functions of \(y\) and \(\frac{dy}{dt}\) on the LHS, you must divide both sides by \((y + 1)\). Simply subtracting \(y\) from both sides does not result in a product of functions of \(y\) on the LHS. It is also important to remember that y is an unknown function. The operation \(\int y\,dt\) makes perfectly good sense. It means “whatever the antiderivative of the function \(y\) is”. However, you aren’t going to a more simple answer than what you have – you don’t know what \(y\) is. The answer is certainly not \(\frac{1}{2}y^2\), as you can check in the case that \(y=\cos(t)\).

  3. Separation of Variables. Assuming we can algebraically separate the ODE into the product of functions of \(y\) on the LHS and only constants on the RHS, we can attempt to find a solution to the ODE by integrating both sides of the equation. We use a trick involving the chain rule to get a usable expression on the LHS of the ODE.

7.4 R Commands

antiD

7.5 Problems & Activities

  1. Start with a reminder of where we’ve been in this block. We’ve studied ODEs of the form \(y^{'} = f(t)\). Explain that today we’ll widen our toolbox by exploring differential equations in the form \(y'=f(t)g(y)\).

  2. Practice identifying whether or not an equation is separable. We won’t intentionally try to trick them by asking them if something difficult is separable, but they should be able to recognize that \[\frac{dy}{dt}=yt-t\] is separable by factoring out at \(t\).

  3. Following the motivating examples in the book, solve a couple of problems using integration by parts. The big thing is to separate the variables, and then let the chain rule save the day when you are faced with integrating something like \(y^3\frac{dy}{dt}\). Even though you don’t know what function \(y\) is, the chain rule tells you that the anti-derivative of \(y^3\frac{dy}{dt}\) is \(\frac14 y^4\).

    1. After a couple of examples, it is completely fine to only write the constant of integration on one side or the other of the ODE.

    2. After a couple of examples, if you want to think of “multiplying both sides by \(dt\)” rather than “canceling \(dt\)” as in the book, that is completely fine.

    3. After a couple of examples, it is completely fine to “cancel” the \(dt\)’s and write \[\int y \frac{dy}{dt}\,dt=\int y\,dy.\]

    4. In this section it is completely fine to write \[\int \frac{1}{y}\,dy = \ln(y).\] When this comes up, we’ll always assume that \(y>0\). Once you do that, then you can omit any sign analysis on terms like \(e^{C_1}\). Just turn \(e^{C_1}=C\), as in the book. Yes, I know perfectly well that it should be \(\ln|y|\) and that then we should deal with cases to turn \(e^{C_1}\) into a constant \(C\) with unrestricted sign. For this class that is more trouble than it is worth. They’ll deal carefully with this in an ODE course.

    5. It is totally fine for students to use antiD to do integration. Disappointing, but fine.

    6. Though we are doing separation of variables, we should focus on autonomous differential equations. After a couple non-autonomous examples, switch over to autonomous.

  4. Have students work on problems from exercises in the section. There are plenty of them. Find both general solutions and specific solutions to IVPs.