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Thread: How to solve differential equations ?

  1. #1 How to solve differential equations ? 
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    I ALWAYS WONDERED HOW TO SOLVE DIFFERENTIAL EQUATIONS .
    PLEASE HELP ME TO SOLVE THE FOLLOWING EQUATION.
    (x3+ 3xy2)dx+(y3+ 3x2y)dy = 0
    A KIND NOTE: This is not my homework or some assignment.So,kindly help me.
    This is out of my interest.


    Last edited by kvskvt; July 11th, 2012 at 11:30 AM.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by kvskvt View Post
    I ALWAYS WONDERED HOW TO SOLVE DIFFERENTIAL EQUATIONS .
    PLEASE HELP ME TO SOLVE THE FOLLOWING EQUATION.
    (x3+ 3xy2)dx+(y3+ 3x2y)dy = 0
    A KIND NOTE: This is not my homework or some assignment.So,kindly help me.
    This is out of my interest.
    The above differential equation would be difficult to solve as it is highly nonlinear; in fact, by looking at its general form, I don't think there even is a solution to this equation in terms of elemental functions. Perhaps it could be done numerically.
    If you want to learn about differential equations you need to start with something much more basic.


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    Forum Senior TheObserver's Avatar
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    This is an exact equation, it shouldn't be hard to solve. You can confirm that by noting that the y derivative of the dx term is equal to the x derivative of the dy term. So you need to find the function whose x derivative is x^3+ 3xy^2 and whose y derivative is y^3+ 3x^2y. By inspection the solution is just (1/4)*x^4 + (1/4)*y^4 + (3/2)(x^2)(y^2). You should have learned a more algorithmic way to do it using integrating factors in a ODE class.
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  5. #4  
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    Quote Originally Posted by kvskvt View Post
    I ALWAYS WONDERED HOW TO SOLVE DIFFERENTIAL EQUATIONS .
    PLEASE HELP ME TO SOLVE THE FOLLOWING EQUATION.
    (x3+ 3xy2)dx+(y3+ 3x2y)dy = 0
    A KIND NOTE: This is not my homework or some assignment.So,kindly help me.
    This is out of my interest.
    I am not at all convinced by your declaration that this isn't some assignment. The differential equation you posed seems quite contrived. It is not what one would expect as a random example, but it is one that would be assigned by some evil professor.

    But now that you've been given the answer (within an integration constant), perhaps we can try to salvage some educational value despite my suspicions. First note that the equation you presented has a nice symmetry property -- if you exchange x and y everywhere, you get the same equation. This exchange invariance is a sign that you should attempt to solve it by the method of "separation of variables." That's all I'll write. Whether or not you got the equation from a homework assignment, you should now search your textbook or online source(s) for how to solve differential equations by this approach. Also pay attention to what limited class of problems is amenable to solution by this method.
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  6. #5  
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    Quote Originally Posted by tk421 View Post
    I am not at all convinced by your declaration that this isn't some assignment. The differential equation you posed seems quite contrived. It is not what one would expect as a random example, but it is one that would be assigned by some evil professor.
    I too find it a bit complicated to find out of interest, and I'm trying to learn differential equations independently.
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    Forum Senior TheObserver's Avatar
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    Quote Originally Posted by tk421 View Post
    It is not what one would expect as a random example, but it is one that would be assigned by some evil professor.
    Is this not just a basic exact equation? Pretty much standard for a first course on ODE, it seems like about as simple as you can get in terms of an example of an exact equation.
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  8. #7  
    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by TheObserver View Post
    This is an exact equation, it shouldn't be hard to solve. You can confirm that by noting that the y derivative of the dx term is equal to the x derivative of the dy term. So you need to find the function whose x derivative is x^3+ 3xy^2 and whose y derivative is y^3+ 3x^2y. By inspection the solution is just (1/4)*x^4 + (1/4)*y^4 + (3/2)(x^2)(y^2). You should have learned a more algorithmic way to do it using integrating factors in a ODE class.
    I think y is not a variable, but the unknown function itself - it should really read y(x). And with this the DE becomes very complicated.
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    this is the form of an exact equation that I am used to from my ODE class. (This was taken from the wiki page on exact equations)

    Seems to be exactly what we have here.
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  10. #9  
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    Quote Originally Posted by TheObserver View Post
    this is the form of an exact equation that I am used to from my ODE class. (This was taken from the wiki page on exact equations)

    Seems to be exactly what we have here.
    Yes, you may be right.
    I took y to really mean y(x), which would make this whole thing a completely different kettle of fish !
    halorealm likes this.
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  11. #10  
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    Quote Originally Posted by Markus Hanke View Post
    Yes, you may be right.
    I took y to really mean y(x), which would make this whole thing a completely different kettle of fish !
    Yeah if those aren't independent variables then I don't really know what to make of this, but I'm sure they are. Its exactly the kind of problem I was doing last semester.
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  12. #11  
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    In fact this problem is probably best thought of as describing a conservative vector field. Then the solution becomes a matter of finding the scaler potential.
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