anonymous
  • anonymous
i need help in finding the solution of this ordinary differential equation 3(3x^2+y^2)dx-2xydy=0 using any method.
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
goformit100
  • goformit100
@yeyenunez "Welcome to OpenStudy. I can answer your questions or guide you. Please use the chat for off topic questions. And remember to give the helper a medal, by clicking on "Best Answer". We follow a code of conduct, ( http://openstudy.com/code-of-conduct ). Please take a moment to read it."
anonymous
  • anonymous
Lets answer :)
anonymous
  • anonymous
@E.ali ok :D

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More answers

Mimi_x3
  • Mimi_x3
\[ 3(3x^2+ y^2)dx - 2xy\frac{dy}{dx}=0\] \[ 9x^2 + 3y^2dx - 2xy\frac{dy}{dx}=0\] \[ -2xy\frac{dy}{dx} + 3y^2dx + 9x^2 =0 => -2xy\frac{dy}{dx} + 3y^2dx = -9x^2\] \[ 2ydy - \frac{3y^2}{x} = 9x\] Let \(u = y^2\) \[ \frac{du}{dx} = 2y\frac{dy}{dx}\] \[ => \frac{du}{dx} - \frac{3u}{x} =9x\] so it's a linear ode where the solution is \[ y = e^{-h(x)} [ \int e^{h(x)rdx +c\]
anonymous
  • anonymous
hint, equation is exact
Mimi_x3
  • Mimi_x3
i dont think so..
anonymous
  • anonymous
ups, you right. Didn't see the - sign
anonymous
  • anonymous
\[3(3x ^{2}+y ^{2})dx-2xydy=0\] \[9x ^{2}+3y ^{2} -2xydy=0\] \[9dx-\frac{ 2xydy-3y ^{2}dx }{ x ^{2} }=0\] There is an integrable combination wherein: \[d(\frac{ y ^{2} }{ x })=\frac{ 2xydy-y ^{2}dx }{ x ^{2} }\] However I am having problems with the \[-3y ^{2}\] in order to apply that integrable combination :)
anonymous
  • anonymous
this is homogeneous ode. Write it like this: \(y'=\huge\frac{2xy}{3x^2+y^2}\) now reduce it single variable by multiplaying up and down part by \(1/x^2\) . Later you can solve it by taking variable change \(u=y/x\)
anonymous
  • anonymous
which makes it separate variable, and simple integration, :)
anonymous
  • anonymous
@Mimi_x3 u'r solution is wrong
Mimi_x3
  • Mimi_x3
how lol?
anonymous
  • anonymous
where that 3 come out from?
Mimi_x3
  • Mimi_x3
the question lol?
anonymous
  • anonymous
@myko uhm its \[3(3x ^{2}+y ^{2})\] so if you get its partial derivatives its not homogeneous and there is also a negative sign O.O
anonymous
  • anonymous
Hey guys ! I have a better way !:)
Mimi_x3
  • Mimi_x3
is the ode exact?
anonymous
  • anonymous
equation is homogenious
anonymous
  • anonymous
and u solve the way i explaned
anonymous
  • anonymous
@: myko : Wait please !
terenzreignz
  • terenzreignz
Bernoulli?
anonymous
  • anonymous
We have : 9x^2+2y^2.dx -2(x-y^2-d)=0 ok ?!
anonymous
  • anonymous
\[f(x,y) = 3(3x ^{2}+y ^{2})dx - 2xydy=0\] \[f(\lambda x,\lambda y) = 3(3(\lambda x)^{2}+(\lambda y)^{2})- 2(\lambda x)(\lambda y) =0\] so it is homogenous lol =))))
anonymous
  • anonymous
And now we can have : 9x^2+2y^2.dx=2(-x+y^2+d) Ok?!
anonymous
  • anonymous
And now :! CAN you complate it ?!
anonymous
  • anonymous
@E.ali can you use the equation button I can't understand your equation...
anonymous
  • anonymous
the idea is to first separate the variables all y's on one side all x's on the other side to accomplish that, let's start by distributing in the first parenthesis go ahead and give that a try please.
anonymous
  • anonymous
@ ; Archie : We dont have just x and y .! We have d to !
anonymous
  • anonymous
distributing means multiplying the factor in front of the parentheses by each and every term inside the parenthesis here is an example: a(x+y) becomes ax + ay that is called distributing the a to the two terms inside the parenthesis.
anonymous
  • anonymous
Sure we do, but we are trying to do this step by step. if you want, you can distribute both the 3 and the dx at the same time
anonymous
  • anonymous
but I suggest you do it step by step
anonymous
  • anonymous
@ⒶArchie☁✪ \[9x ^{2}dx+3y ^{2}dx=2xy dy\] \[9x ^{2}dx=2xy dy-3y ^{2}dx\]
anonymous
  • anonymous
am i doing it right?
anonymous
  • anonymous
Perfect :)
anonymous
  • anonymous
but you don't need to move the 3y^2dx term over to the right let's keep that on the left, together with the other term that has dx in it have a look:
anonymous
  • anonymous
|dw:1377938975495:dw|
anonymous
  • anonymous
alright what do we need to do next, to only have y's on the right hand side?
anonymous
  • anonymous
It s not important !
anonymous
  • anonymous
if it is helping yeyenunez learn, then I definitely think it is important E.ali.
anonymous
  • anonymous
also yes it is important we want the variables separated before we can integrate each side
Callisto
  • Callisto
\[3(3x^2+y^2)dx-2xydy=0 \]\[3(3x^2+y^2)dx=2xydy\]\[\frac{dy}{dx}=\frac{9x^2+3y^2}{2xy}\]\[\frac{dy}{dx}=\frac{9x}{2y}+\frac{3y}{2x}\]Let y=wx \[\frac{dy}{dx}=w+\frac{dw}{dx}x\] Then, the DE becomes \[w+w'x = \frac{9}{2w}+\frac{3}{2}w\]\[w'x=\frac{9}{2w}+\frac{w}{2}\]\[w'x=\frac{9+w^2}{2w}\]\[w'x=\frac{9}{2w}+\frac{w}{2}\]\[\frac{dw}{\frac{9+w^2}{2w}}=\frac{dx}{x} \]
anonymous
  • anonymous
@ Archie : We have : 9x^2 + 3y ^2=2xy^2 Ha?!
anonymous
  • anonymous
@ⒶArchie☁✪ |dw:1377939104234:dw|?
anonymous
  • anonymous
Exacly !
anonymous
  • anonymous
My way is such as your way !
anonymous
  • anonymous
right
anonymous
  • anonymous
Then use Radical !
anonymous
  • anonymous
@E.ali i think you are not helping me at all :( I'm having problems with your equations because i cannot understand it at all
anonymous
  • anonymous
It s 9x + 3y =2xy !@yeyenunez:Ok?!
anonymous
  • anonymous
@yeyenunez Can we start fresh here? There is a better way to solve this equation.
anonymous
  • anonymous
@ⒶArchie☁✪ ok
anonymous
  • anonymous
|dw:1377939464495:dw|
anonymous
  • anonymous
I've written the original DE on the board. A quick bit of theory next! :) Suppose we have a function F(x,y) and look at its differential:
anonymous
  • anonymous
@E.ali you cannot use radicals on terms with a plus sign in between ~_~ so 9x^2+3y^2 is not equal to 9x +3y ~_~
anonymous
  • anonymous
LOOK ! WE can Reduse from all . We can have Radical from all ... and ...
anonymous
  • anonymous
|dw:1377939552415:dw|
anonymous
  • anonymous
So the differential dF can be expressed as this sum that involves partials of F with respect to X and Y. I trust your teacher has covered this type of differential expression.
anonymous
  • anonymous
Are you familiar with partial derivatives? @yeyenunez
terenzreignz
  • terenzreignz
@yeyenunez As you have already found and pointed out, the equation is homogeneous... so why don't you start with letting y = vx and working from there? ^_^
anonymous
  • anonymous
This expression for dF is the general formula for a function of 2 variables Just want to make sure you are ok with this general formula for dF and know about partial derivatives.
anonymous
  • anonymous
@ⒶArchie☁✪ we are only allowed to use partial derivatives when proving exactness of ode's
anonymous
  • anonymous
This is what we have here with this DE. oh okay then, well I suggest you to look at the previous posts given by other people and go on from there.
anonymous
  • anonymous
@ⒶArchie☁✪ thanks for the help :)
anonymous
  • anonymous
Your welcome, good luck with your studies. [:
anonymous
  • anonymous
@terenzreignz wait i'll try using the homogenoeus method
anonymous
  • anonymous
@Callisto what type of method did you use?
Callisto
  • Callisto
Homogeneous.
anonymous
  • anonymous
@Callisto our homogenous method is quite different from yours... i don't think our professor would accept that kind of solution but still thanks for you help :)
Callisto
  • Callisto
What's the difference?
anonymous
  • anonymous
@yeyenunez perhaps you would like to share your homogenous method with us? [:
anonymous
  • anonymous
we use the y=vx and our equation must be in the form Pdx+Qdy=0
terenzreignz
  • terenzreignz
@yeyenunez out of interest, who IS your professor? (UPD kid here ^_^)
Callisto
  • Callisto
It is just the same. I've just rearranged the terms.
anonymous
  • anonymous
@terenzreignz sir sigua =)))))
terenzreignz
  • terenzreignz
Neeever heard of him. Anyway, just let y = vx and the rest falls into place. What Calli did isn't that much different, she just delayed the substitution until a bit later into the solution.

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