I was having trouble with this question as I'm not sure how to split it into dy and dx in order to integrate.
x*dy/dx+y = x given y(-1)=2.
y= ?
Thanks, Daddy_Long_Legs
$\displaystyle y'+\frac{y}{x}=1$
find your integrating factor...
$\displaystyle \mu{(x)}=e^{\int{\frac{1}{x}}dx}\Rightarrow\mu{(x) }=x$
Multiply both sides by your integrating factor and integrate...
$\displaystyle \int{(yx)'}=\int{x}$
$\displaystyle yx=\frac{x^{2}}{2}+c$
Solve for C
$\displaystyle 2(-1)=\frac{(-1)^{2}}{2}+C$
$\displaystyle C=-\frac{5}{2}$
Now, solve for y ($\displaystyle f(x)$) and you are done.
$\displaystyle f(x)=\frac{x}{2}-\frac{5}{2x}$
Seems like you have never been introduced to this method, so check out this wiki page: Integrating factor - Wikipedia, the free encyclopedia
It's pretty straight forward, just takes some practice.
Danneedshelp,
I've read this link and I'm still stuck with this question, I have followed your working but I'm afraid I can't see how you got the answer. I was wondering if you or another member could show me again as this question is coming up repeatedly and I'm still clueless on to do an intergrading factor.
Thanks.
Hi Daddy_Long_Legs
Maybe this will be helpful^^
http://www.mathhelpforum.com/math-he...-tutorial.html
songoku and Danneedshelp,
I have come across this question a few times in my reviews and I have entered this answer several times and I keep getting the question wrong and I'm not sure why? The question mentions parenthesizing the denominator carefully but I have two dominators which leads me to believe that the answer I have been entering is incorrect or that it needs rearranging of some kind.
Thanks.
A perfectly good answer, already given, is $\displaystyle y = \frac{x}{2} - \frac{5}{2x}$.
I'll bet that whoever gave you this question wants the answer written as $\displaystyle y = \frac{x^2 - 5}{2x}$, which is completely pointless since the point of the question, I would have thought, is to test whether you can solve a DE, not whether you can read the (narrow and small) mind of someone.
(A likely scenario is that the person who gave you this question has copied it from somewhere and is slavishly using the given answer rather than exercising some common sense and/or intelligence).