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Classify this diff eq

  1. Oct 5, 2004 #1
    x(dy/dx) = y*e^(x/y) - x

    its either separable, linear, homogeneous, bernoulli or exact. only thing i can figure is that its linear.

    how do i break it down to figure it out? the e^x/y is whats throwing me off.
     
  2. jcsd
  3. Oct 5, 2004 #2

    arildno

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    It's not linear!!
    Divide by x.
    Note that your right-hand side can now be written as some function g(y/x).
     
  4. Oct 5, 2004 #3
    if i break it apart i get dy/dx = (y*e^(x/y) / x) - 1. i can see how it could be separable, but the e^x/y would still be there when i integrate. and that integral would be fairly impossible. i dont see what else it can be.
     
  5. Oct 5, 2004 #4

    arildno

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    Introduce the variable:
    [tex]v(x)=\frac{y(x)}{x}[/tex]
    We have then:
    [tex]v'(x)=\frac{y'(x)}{x}-\frac{v}{x}[/tex]
    Or:
    [tex]y'(x)=xv'(x)+v(x)[/tex]
    Hence, inserting this into your diff. eq., you have:
    [tex]xv'(x)=ve^{\frac{1}{v}}-(1+v)[/tex]
    This is a separable equation (I wouldn't try solving it, though..)
     
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