Math 391 Lecture 6 - Exact Equations and Reduction of Order

In this lecture, we complete the topics required for the first test--Section 2.6 Exact Equations, and the miscellaneous section of 2.9 where we look at the Reduction of Order technique (we will look at even more techniques next time). We define what an exact differential equation is, how to tell when an equation is exact and how to find the solution to an ODE given that it is exact. We cover two methods of finding a solution to an exact equation, and do a few examples.

We end by doing one example of a reduction of order problem--where we used a substitution to change a second order ODE into a first order ODE, and hence solved it. We stop at only one example, since in this context the technique is straight forward, and we shall revisit reduction of order in a later chapter, in a more general context.

[muazozpolat]

Professor Jhevon, just wanna say thank you for saving my life with this course haha. I have a difficult time understanding the way the professors at my university teach diff, your videos are extremely beneficial and go hand in hand with our syllabus. Again, thank you. METU student

[qucikquestion]

At 34:50 is it allowed to multiply through by dx, Ive seen some sources do it while others say it is not mathematically rigorous. What is the mathematically rigorous way. Ive been having trouble conceptually understanding differentials as in some scenarios they are treated as fractions while others they say it is just convenient notation.
Side note: Great videos, great teaching Thank you for posting them they are incredibly helpful you give MIT OCW a run for their money.

[brockwin]

It does not seem that you sufficiently explained why method 2 works?

[maguiar]

I loved that method for solving integration by parts at 43:00. Sadly it won't work if both of them, u and v', are infinitely differentiable. Thanks a lot for the tips, Jhevon! :)

[UnforsakenXII]

Is there ever a situation where the mixed partial derivatives do not match up? If so, is that what we call inexact equations? In thermodynamics, there's a different notation used to depict the path dependence of a path function contrary to the usual functions that we use.

For example, to prove that a certain quantity was path independent, they made me use a "line integral" (still not sure if it's used in math) in order to integrate over a closed path and then see if the quantity is zero and they mentioned that it had some correlation with second order mixed partials but they were really hand-wavy about it.

Is there a class in which you learn about these inexact differentials?

Actually you might answer it later in the vid, i'm about at 15:00. hmm

[manuelsanchez]

could you explain why the product rule can be used here? @57:01