Question in understanding differitial equations

  • Thread starter transgalactic
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In summary, the differential equation x \ dy -y \ dx = 0 is not exact, but it can be made to be exact by multiplying throughout by the 1/x^2:
  • #1
transgalactic
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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

?
 
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  • #2
It looks very much like an http://mathworld.wolfram.com/ExactDifferential.html" to me.
 
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  • #3
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.
 
  • #4
i too see it as an exact differential
it looks like the one in the article

why its not an exact differential??
 
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  • #5
They differ by a factor of -1.
 
  • #6
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?
 
  • #7
I did this: [tex]\frac{\partial}{\partial y} \ \frac{y}{x} = \frac{1}{x}[/tex]
[tex]\frac{\partial}{\partial x}\ \left(y^2-lnx\right) = -\frac{1}{x}[/tex].

That's why I said they differ by factor of -1.
 
  • #8
what does the expression d/dx represent

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

?
 
  • #9
Uh, which expression? This one: [tex]\frac{\partial}{\partial x} [/tex]? That's partial differentiation. It means to say differentiate a multi-variable function with respect to x alone.
 
  • #11
Actually I don't know what you're writing:

Specifically, what does [tex]d(ln|y|) \ \mbox{and} \ d(ln|x|)[/tex] mean?
 
  • #12
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
?
 
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  • #13
I really have no idea what you are saying. What patterns of integrals are you talking about?
 
  • #14
i/x i recognized as the derivative of ln|x|

how to solve this correctly
 
  • #15
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 [tex]N(x,y) \ dy + \ M(x,y) \ dx =0[/tex] is considered exact if [tex]\frac{\partial M}{\partial y} \ = \ \frac{\partial N}{\partial x}[/tex].

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

[tex]\frac{1}{x} \ dy - \frac{y}{x^2} \ dx = 0[/tex]

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

What is a differential equation?

A differential equation is an equation that involves an unknown function and its derivatives. It describes the relationship between the function and its derivatives, and is commonly used to model change and rates of change in various scientific fields.

What is the difference between ordinary and partial differential equations?

An ordinary differential equation involves only one independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are used to model systems that change over time, while partial differential equations are used to model systems that change over space or other dimensions.

Why are differential equations important?

Differential equations are important because they provide a mathematical framework for understanding and predicting the behavior of complex systems. They are used to model a wide range of phenomena in fields such as physics, engineering, economics, and biology.

What are the applications of differential equations?

Differential equations have many applications in science and engineering. They are used to model physical systems such as the movement of objects, the flow of fluids, and the behavior of circuits. They are also used in fields such as economics, population dynamics, and epidemiology.

What are some techniques for solving differential equations?

There are several techniques for solving differential equations, including separation of variables, integrating factors, and using series solutions. Other methods such as numerical methods and Laplace transforms can also be used to solve differential equations. The most appropriate technique depends on the specific type of differential equation and its complexity.

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