Solve the differential equation y'+2y/(x+4)=(x+4)^2 where y=2 when x=0. y(x)= ?

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Solve the differential equation y'+2y/(x+4)=(x+4)^2 where y=2 when x=0. y(x)= ?

Mathematics
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You can use the method of integrating factors. Do you know how to do that?
\[y'+\frac{2}{x+4}y=(x+4)^2\]
Oh i haven't learned that

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Oh - so, do you want me to just go through the motions and you look up the method?
yea sure
We look for a function mu(x) such that,\[\frac{d (\mu y)}{dx}=\mu (x+4)^2\]
μ=2/(x+4) in this case?
This is one of those things I usually derive for myself each time (since there's less chance of stuffing up), but for an equation in the form\[y'+p(x)y=q(x)\]the integrating factor is\[\mu(x)=e^{\int\limits_{}{}p(x)dx}\]
So here,\[\mu = \exp (\int\limits \frac{2}{x+4}dx)=e^{2\log (x+4)}=e^{\log (x+4)^2}=(x+4)^2\]
You then sub in mu, and integrate both sides:\[\frac{d((x+4)^2y)}{dx}=(x+4)^2(x+4)^2\]
\[\rightarrow (x+4)^2y=\int\limits_{}{}(x+4)^4dx\]\[=\frac{1}{5}(x+4)^5+c\]That is,\[(x+4)^2y=\frac{1}{5}(x+4)^5 +c \rightarrow y = \frac{1}{5} (x+4)^3+\frac{c}{(x+4)^2}\]
I just checked with Wolfram and it's right.
I got everything besides those formulas haha
You mean, why we set the problem up like that?
yes
Yeah, that's why I asked if you were familiar with the method, because I knew it wouldn't make sense (looks like you're pulling something out of nowhere).
Go to this page, http://www.khanacademy.org/ and look up 'integrating factors' under the differential equations section.
this is a math homework problem from my friend who is taking a different series of math courses here
oh khanacademy. i watch their videos quite often on youtube :)
I'm just sketching a proof...
I'll scan.
thanks for all this hot stuff :)
are you a math teacher ?
I tutor tertiary and secondary maths to pay for postgrad.
Oh
In (1) we just multiply through by some function, mu(x). We know nothing about it yet. Then we put it aside and consider d(mu y)/dx. Then we link the two together to force a mu that has the property we want.
mu in mu(x) is just a symbol like f as in f(x)
yes
\[\mu = \mu (x)\]
it makes sense :)
Good :)
Can you explain partial fractions? What i dont understand is that why f(x)/g(x) is equal to A1/(x-r)+A2/(x-r)^2+A3/(x-r)^3+.......Am/(x-r)^m if g(x) is a product of linear factors(x-r)^m Why doesn't the degree of the numerators increase as the degree of the denominator increases, but rather stays as constants A1....Am?
The theory behind it is that you have repeated roots in the polynomial g(x). The method has to be modified to take account of the repeating of the roots.
I think you're asking this because you see the situation where there is a quadratic in the denominator sometimes, and you're told to write the numerator as Ax+B, say. You're looking at (x-a)^m and thinking, "This is a polynomial of degree m, so why aren't I using a polynomial of degree m-1 in my numerator?" Is that what it is?
Yea exactly
If so, it's because the quadratics (and higher polynomials) in those cases are irreducible over the field we're working in. For example, in the real field, \[x^2+x+1\]has no real factors. So over the real field, this thing is *irreducible*: we cannot write it as a product of linear factors. We have to then use a numerator that has a polynomial of degree one less. In the case where g(x)=(x-a)^m, if you saw it in expanded form, although you wouldn't necessarily be able to eyeball it, it *is* reducible into linear factors. That's all, really.
for instance, if \[f(x)\div g(x) = (3x)\div (x+1)^3 -> A \div (x+1)+B \div (x+1)^2+ C \div (x+1)^3 \] the degree of denominator is increasing, but the degree of the numerator is not. which seems to contradict with Ax+b/ (x^2+x+c)
OH
I see what you mean then should not it be like \[A \div (x+1) + Bx+c \div (x+1)^2\]
The actual theory requires a study of modern algebra. You start talking about polynomials over fields and all sorts of things. The procedure is easy and necessary to do a lot of practical problems, which is why 'method' is taught over 'reason'.
Well, it depends on your original fraction.
Oh I see it.
No, you have *linear* factors here, so you'd write\[\frac{3x}{(x+1)^3}=\frac{A}{(x+1)}+\frac{B}{(x+1)^2}+\frac{C}{(x+1)^3}\]
If you had something like\[\frac{3x}{(x+1)^2(x^2+x+1)}\]then you'd write\[\frac{A}{x+1}+\frac{B}{(x+1)^2}+\frac{Cx+D}{x^2+x+1}\]
You have a repeated linear factor and ONE irreducible quadratic. If the quadratic could be factored, THEN you'd have more LINEAR factors.
because (x+1)^2 is reducible . is that why the numerator there is B but not Bx+C?
It's because you have a linear factor.
See, the whole point behind the procedure is to split up your original polynomial fraction into the sum of the most simplest fractions available.
Why we actually do what we do is, like I said, covered in a modern algebra course.
I like to discover what is behind the methods rather than just plugging numbers into formulas haha it is ok if it takes too much effort to explain :)
When you come across your problem, you consider these things when deciding on the decomposition: 1) are any of my factors repeated? 2) are any of my polynomials irreducible (you're really only ever going to see quadratics in a course you do)? When setting up the decomp., you ALWAYS use a polynomial in the numerator that is 1 degree LESS than the polynomial in the denominator. If your polynomial is all raised to some power, you have to go through the repeating process; e.g.\[\frac{3x}{(x^2+x+1)^2}=\frac{Ax+B}{x^2+x+1}+\frac{Cx+D}{(x^2+x+1)^2}\]
Yeah, like I said, you have to do a course in modern algebra. There's nothing stopping you looking at lecture notes or something. MIT Open Courses sometimes have good material.
Oh i watched most of their single variable calculus videos which are really helpful
Yeah, just keep doing that. Then you won't need this site!
Have you watched the video where the professor used block-stacking to explain convergence and divergence?
No...
LOL my brain is kinda slow.. The more i read, the harder it is for me to think out of box
Oh i watched it but i don't get it :(
Just keep mulling over it. You have to let your head put it together in its own time.
http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/lec36.pdf Do you explain how he gets the equation on page 7?
do you mind explaining*
I could explain it if I had more time...I have to get ready for uni/work. I will definitely look at it and get back to you.
Thanks :)
Later ;)
I had a look at your question during the day. I'll write something up for you soon. The lecturer just made some assumptions in the end that they didn't make explicit, mainly because (I'm assuming) that part of the lecture wasn't core.
I watched the lecture video too, i don't get why \[C_{n+1}=C_{n}+1 \] he says the center of mass of the N+1 block(the last block) is 1 unit further than the center of mass of the N blocks
It doesn't. \[C_{n+1}=C_n+\frac{1}{n+1}\]
Dichalao, I will be back in a min. Stay here.
How you found it from that piles of questions
E-mail
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Can you see the picture i posted?
yes
the professor said becasue the length of the block is 2, that is why C_(n+1)=Cn+1
Sorry, this thing keeps freezing up.
No problem :)
Yeah, the center of the (n+1)st block is at C_n+1 because you're measuring horizontally from the tip of the first block on top to the center of the (n+1)st block. I'll draw it.
And each block has 2 units of length - don't forget that.
and the center of mass of each block individually is in the geometrical center in this situation.
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I added the geometry.
You're just dealing in distance. The C_n is the collective distance up to the beginning of the (n+1)st block and the 1 comes from the 2 units of block. He's not talking about the center of mass, just geometric center I think.
The notes are what you should be paying attention to. They make complete sense.
C_1 is one unit away from the origin, C_2=1+1/2, C_3=1+1/2+1/3,, then when it comes to C_(n+1)=Cn+1 because every succeeding term is getting bigger and bigger, but the component added to the previous term is getting smaller and smaller
Yes, basically. If you look at the notes, you'll see that C_n is the collective distance (hence why I think he's used 'C') from the beginning of the first block up to the center of mass of the nth block. The physics is what determines these distances. The mathematics is what determines the solution. Here, if you look, you have C_0 = 0 (because it's at the tip, and therefore no distance) C_1 = C_0 + 1 (because the center of mass of a block is in its geometric center) C_2 = C_0 + C_1 +1/2 (because the physics says this is where the CoM is and we're aiming to place the center of mass at the edge of the block below) C_3=C_0+C_1+C_2+1/3 (because of what I just said for C_2) and so it continues...
You build up a series,\[C_N=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{N}\]
Following so far
for the distance from the beginning to the center of mass of the Nth block.
His point in this exercise is to show that this physical system can be set up in theory AND that you can make its horizontal projection (i.e. how the blocks extend in the x-direction, say) as large as you want (people would tend to think it would be impossible). He shows you can make it as large as you want BECAUSE the (collective) distance is \[C_N=\sum_{n=0}^{N}\frac{1}{n}\]...a harmonic series. These things DON'T converge, so as n gets larger, so does the distance stretched...that's his point.
But at the same time, the system has been set up to be stable.
I got what you saying here
So you're fine with it?
Sure :) Thanks~~
No probs. :)

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