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Differential Equations and Substitutions (Calc 2)

  1. Oct 10, 2008 #1
    1. The problem statement, all variables and given/known data
    Solve xy' = y + xe^(y/x) using the substitution v=(y/x)


    2. Relevant equations
    Solving differential equations, substitution


    3. The attempt at a solution
    x (dy/dx) = y + xe^(y/x)

    (dy/dx) = (y/x) + e^(y/x)

    Substituting v=(y/x)

    (dy/dx) = v + e^(v)

    I do not know how to proceed from here. (There are so many variables that aren't x and y! Ahh) Any guidance would be greatly appreciated!
     
  2. jcsd
  3. Oct 10, 2008 #2

    Hootenanny

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    You need to make the change of variable with the differential too. In other words you need to write

    [tex]\frac{dy}{dx}[/tex]

    In terms of x and v. Note that v=v(x), that is, v is a function of x.
     
  4. Oct 10, 2008 #3

    HallsofIvy

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    If v=y/x, then y= ??

    and from that dy/dx= ??
     
  5. Oct 10, 2008 #4
    Hmm, ok, so dv = dy/dx? Somehow I still think I'm missing something. Shouldn't there be a dv/dx somewhere or something? I am just not seeing it =/
     
  6. Oct 11, 2008 #5

    Hootenanny

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    You're on the right lines, but not quite there. As Halls says,

    [tex]y = v(x)\cdot x[/tex]

    Now you need to find the first derivative of the above function with respect to x,

    [tex]\frac{dy}{dx} =\ldots[/tex]
     
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