Question

i need help in finding the solution of this ordinary differential equation 3(3x^2+y^2)dx-2xydy=0 using any method.

Answer 1

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Answer 2

Lets answer :)

Answer 3

@E.ali ok :D

Answer 4

\[ 3(3x^2+ y^2)dx - 2xy\frac{dy}{dx}=0\]
\[ 9x^2 + 3y^2dx - 2xy\frac{dy}{dx}=0\]
\[ -2xy\frac{dy}{dx} + 3y^2dx + 9x^2 =0 => -2xy\frac{dy}{dx} + 3y^2dx = -9x^2\]
\[ 2ydy - \frac{3y^2}{x} = 9x\]
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
\[ y = e^{-h(x)} [ \int e^{h(x)rdx +c\]

Answer 5

hint, equation is exact

Answer 6

i dont think so..

Answer 7

ups, you right. Didn't see the - sign

Answer 8

\[3(3x ^{2}+y ^{2})dx-2xydy=0\]
\[9x ^{2}+3y ^{2} -2xydy=0\]
\[9dx-\frac{ 2xydy-3y ^{2}dx }{ x ^{2} }=0\]
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
:)

Answer 9

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 \(u=y/x\)

Answer 10

which makes it separate variable, and simple integration, :)

Answer 11

@Mimi_x3 u'r solution is wrong

Answer 12

how lol?

Answer 13

where that 3 come out from?

Answer 14

the question lol?

Answer 15

@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

Answer 16

Hey guys !
I have a better way !:)

Answer 17

is the ode exact?

Answer 18

equation is homogenious

Answer 19

and u solve the way i explaned

Answer 20

@: myko : Wait please !

Answer 21

Bernoulli?

Answer 22

We have :
9x^2+2y^2.dx -2(x-y^2-d)=0
ok ?!

Answer 23

\[f(x,y) = 3(3x ^{2}+y ^{2})dx - 2xydy=0\]
\[f(\lambda x,\lambda y) = 3(3(\lambda x)^{2}+(\lambda y)^{2})- 2(\lambda x)(\lambda y) =0\]
so it is homogenous lol =))))

Answer 24

And now we can have :
9x^2+2y^2.dx=2(-x+y^2+d)
Ok?!

Answer 25

And now :!
CAN you complate it ?!

Answer 26

@E.ali can you use the equation button I can't understand your equation...

Answer 27

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.

Answer 28

@ ; Archie :
We dont have just x and y .!
We have d to !

Answer 29

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.

Answer 30

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

Answer 31

but I suggest you do it step by step

Answer 32

@ⒶArchie☁✪
\[9x ^{2}dx+3y ^{2}dx=2xy dy\]
\[9x ^{2}dx=2xy dy-3y ^{2}dx\]

Answer 33

am i doing it right?

Answer 34

Perfect :)

Answer 35

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: