Toronto Math Forum

MAT244--2018F => MAT244--Lectures & Home Assignments => Topic started by: Weina Zhu on September 19, 2018, 03:41:41 PM

Title: some inspiration for integration factor
Post by: Weina Zhu on September 19, 2018, 03:41:41 PM
I found a method online suggested by a youtuber Michel van Biezen.
Now we have a non-exact question, we multiply $x^ny^m$ both side to the homogeneous equation.
Then we want the new $M_y=N_x$, the question becomes to find appropriate $m$ and $n$, that will be easier since we do not need to guess the dependence of ....

Hope this helps.

reference:https://www.youtube.com/watch?v=xt6Gw3rzMwI (https://www.youtube.com/watch?v=xt6Gw3rzMwI)
Title: Re: some inspiration for integration factor
Post by: Victor Ivrii on September 19, 2018, 06:13:55 PM
It does not really help. Usually integrating factor cannot be found in any reasonable way.

Sometimes, indeed, integrating factor, could be found in the form $\mu(x,y)=x^m y^n$ but you need to guess that it could be found in this form. Sometimes integrating factor  could be found in the form $\mu(x,y)=\mu(x)$, or $\mu(x,y)=\mu(y)$, or $\mu(x,y)=\mu(xy)$. So advice of this "tuber" is as good as what is written in the textbooks and in the lectures? No, and there are two reasons for this

1. There are "much more" functions $\mu=\mu(x)$ than $\mu =x^my^n$.
2. Problems are created by the authors and instructors to match certain recipes, and problems in most of the textbooks (including the one we use), our home assignments, quizzes, tests, final exam match  to $\mu(x,y)=\mu(x)$, or $\mu(x,y)=\mu(y)$, or $\mu(x,y)=\mu(xy)$. They are not specifically designed to match $\mu =x^my^n$. It may happen occasionally but only occasionally.

On the other hand, if you were taking the class of this "tuber", then the problems would be designed to match $\mu =x^my^n$. So his advice works reliably in his class and his class only. In the rest of the known Universe it is a bad advice.

You can find a lot of things on the internet, but mostly garbage. Do not bring it here.

PS I have not deleted your post and spent some time replying to avoid a conspiracy theory that instructors hide the truth from the students 🤣