Question in understanding differitial equations

[transgalactic]

i have started to learn this stuff
and it looks like a normal derivative stuff but its different
i don't know how to find a general solution
what is the algorithm to act on??

here is a simple example:
http://img381.imageshack.us/my.php?image=26202865up1.jpg
i was told to find a general solution

i don't know how to isolate Y i am not sure
what i need to do here
its so much different

?

 

[Hootenanny]

It looks very much like an http://mathworld.wolfram.com/ExactDifferential.html" to me.

 

[Defennder]

No, it's not an exact differential equation... yet. You have to multiply the entire expression throughout by something to make it exact, then you can solve it by that method.

 

[transgalactic]

i too see it as an exact differential
it looks like the one in the article

why its not an exact differential??

 

[Defennder]

They differ by a factor of -1.

 

[transgalactic]

how did you get -1??

there are two variables here
there could be also derivatives in one of them

how did you get the factor?

 

[Defennder]

I did this: $$\frac{\partial}{\partial y} \ \frac{y}{x} = \frac{1}{x}$$
$$\frac{\partial}{\partial x}\ \left(y^2-lnx\right) = -\frac{1}{x}$$.

That's why I said they differ by factor of -1.

 

[transgalactic]

what does the expression d/dx represent

when i did derivatives there were only dx not "d"

?

 

[Defennder]

Uh, which expression? This one: $$\frac{\partial}{\partial x}$$? That's partial differentiation. It means to say differentiate a multi-variable function with respect to x alone.

 

[transgalactic]

i tried to solve it

http://img405.imageshack.us/my.php?image=img8853jl1.jpg

but i got stuck
what to do next??

 

[Defennder]

Actually I don't know what you're writing:

Specifically, what does $$d(ln|y|) \ \mbox{and} \ d(ln|x|)$$ mean?

 

[transgalactic]

i recognized 1/x and 1/y as a derivatives of ln|x| and ln|y|

thats the only integrals i saw there

i read that i am supposed to see patterns of integrals
and solve them

but its not working here
?

 

[Defennder]

I really have no idea what you are saying. What patterns of integrals are you talking about?

 

[transgalactic]

i/x i recognized as the derivative of ln|x|

how to solve this correctly

 

[Defennder]

I don't know how that is related to this question. I'll just refer you to these notes, then maybe you can apply the technique here.

A differential equation of the form $$N(x,y) \ dy + \ M(x,y) \ dx =0$$ is considered exact if $$\frac{\partial M}{\partial y} \ = \ \frac{\partial N}{\partial x}$$.

Eg. $$x \ dy -y \ dx = 0$$ is not exact. But it can be made to be exact by multiplying throughout by the 1/x^2:

$$\frac{1}{x} \ dy - \frac{y}{x^2} \ dx = 0$$

See here for more details on the topic:
http://tutorial.math.lamar.edu/Classes/DE/Exact.aspx