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Questions

abcgo @abcgo

September 2022 1 5 Report

Variable Separable in Differential Equations
[tex]dx+xydy=y^{2}dx+ ydy[/tex] when [tex]x=3[/tex] and [tex]y =0[/tex]

Answers & Comments

kharmsorz

Solution:

$dx+xy\;dy=y^2\;dx+y\;dy\\\Rightarrow xy\;dy-y\;dy=y^2\;dx-dx\\\Rightarrow y(x-1)\;dy=(y^2-1)\;dx\\\\\Rightarrow \dfrac{y}{y^2-1} \;dy=\dfrac{1}{x-1} \;dx\\\\\Rightarrow \displaystyle\int\dfrac{y}{y^2-1} \;dy=\displaystyle\int\dfrac{1}{x-1} \;dx$

$\hspace{25pt}\quad\textsf{where}\displaystyle\int\dfrac{y}{y^2-1}\;dy= \displaystyle\int\dfrac{1}{2u}\;du\\\\\quad\textsf{(Let }u=y^2-1}\quad\Rightarrow du=2y\;dy\quad\textsf{ or }\quad\dfrac{1}{2}du=y\;dy }\textsf{)}\\\\\textsf{}\hspace{115pt}=\dfrac{1}{2} \displaystyle\int\dfrac{1}{u}\;du\\\\\text{}\hspace{115pt}=\dfrac{1}{2}\ln|u|\\\\\text{}\hspace{115pt}=\dfrac{1}{2}\ln|y^2-1|$

$\hspace{25pt}\quad\textsf{where}\displaystyle\int\dfrac{1}{x-1}\;dx= \displaystyle\int\dfrac{1}{u}\;du\\\\\quad\textsf{(Let }u=x-1}\quad\Rightarrow du=dx\textsf{)}\\\text{}\hspace{25pt}\hspace{87pt}=\ln|u|\\\text{}\hspace{25pt}\hspace{87pt}=\ln|x-1|$

$\Rightarrow \dfrac{1}{2}\ln|y^2-1| =\ln|x-1|+c\\\\\Rightarrow \ln|y^2-1| =2\ln|x-1|+C\quad\rightarrow C=2c\\\Rightarrow \ln|y^2-1| =\ln|(x-1)^2|+C\\\Rightarrow \ln|y^2-1| -\ln|(x-1)^2|=C\\\\\Rightarrow \ln\left|\dfrac{y^2-1}{(x-1)^2}\right|=C\\\Rightarrow e^{\ln\left|\dfrac{y^2-1}{(x-1)^2}\right|} =e^C\\\\\Rightarrow \dfrac{y^2-1}{(x-1)^2}=C\quad\rightarrow C=e^C\\\\\Rightarrow \boxed{y^2-1=C(x-1)^2}\quad\leftarrow \textsf{General solution}$

$\textsf{Solve for }C\textsf{ given }x=3,y=0:$

$0^2-1=C(3-1)^2\\\Rightarrow -1=C(2)^2\\\Rightarrow -1=C(4)\\\\\Rightarrow C=-\dfrac{1}{4}$

Answer:

$\Rightarrow \boxed{y^2-1=-\dfrac{1}{4} (x-1)^2}\quad\leftarrow \textsf{Actual solution}$

abcgo For the Variable Separable, before the day ends 11:59 PM haha

abcgo The Non Exact DE is for this coming monday :)

kharmsorz That sounds good. There was a guy who sought help in brainly last night. I was shocked because it was a quiz. So I had to do it quickly. I was sorry for him that I couldn’t help to give all answers, because the time was about to end, and few minutes to take to upload. For me brainly is not good for that kind of situation because the communication wasn’t smooth, you know.

kharmsorz Ok I see. You mean, this is a pre-test exercise, right?

abcgo Yes this is a pre-test. Well I can't really ask for help in an instant for a quiz here haha.

kharmsorz Exactly, unless there was a vöice call or something convenient to talk with, rather than more on to chät.

abcgo There's a message inbox here but I guess you can't really use it unless you answered a lot of questions.

kharmsorz I think, aside from answering a lot of questions, you have to be a Moderator to access the message. Based on my experience, to do that, you must receive many “brainliest” rewards, and you will be given soon a request from a Moderator from which you will find her message in your inbox, asking permission if you’re willing to be a Moderator. In my case, I didn’t care.

kharmsorz Way back few years ago, before 2015 as far as I remember, when my brainly account was in US domain, I was able to message strangers and found it interesting. But because of inappropriate usage brainly had changed their policy. I just read their new policy.

abcgo But to get that you'll be needing stars. Other people doesn't bother giving thank yous and stars after answering the question haha