2 Attachment(s)

Another "mixed partials" question

The question and my work are both attached as PDF files.

Basically, in my work I only found the partial derivatives to be different and therefore did not proceed because of this but apparently this is wrong. Could someone help me figure out what I am doing wrong?

Any help would be greatly appreciated!

Thanks in advance!

Re: Another "mixed partials" question

This time, your should have been -3. You were asked to find an integrating factor **in order to make it** exact. It doesn't appear as though you did that. I'd suggest an integrating factor of the form You might ask how I found that. I started by assuming an integrating factor of the form and then using the exactness condition to solve a differential equation for

Re: Another "mixed partials" question

Could you please elaborate about how I find the an integrating factor to make equations exact because I am breaking my head with this and I am still confused with what you said (I've been rereading it and watching YouTube videos, etc)? Does your way always work? (Assuming that an integrating factor does exist).

Assuming, I followed the procedure correctly, the method here: Integrating factors 1 - YouTube did not work for me for this problem by the way.

Re: Another "mixed partials" question

Here's the basic idea, and this will work for a wide variety of first-order ODE's, but certainly not all first-order ODE's. You multiply through the DE by and then assert the exactness condition. In your case, you have

or

and hence

Asserting the exactness condition yields that

So you whittle things down and simplify, etc., etc., etc. You will often, at some point, have an option to choose one of the exponents, or , in order to simplify things greatly. That's a bit of an art. The three options I would look at first are n=0, m=0, or n=m. Your goal is to solve this first-order differential equation for Then that's your integrating factor.