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Simple ODE, cannot figure out a Way to solve

  1. Nov 5, 2005 #1
    I'm a HS senior, and have never covered differential equations outside a CalculusII curriculum. :frown: This is not homework, rather only a part (just one part!) of an interesting problem I posed myself (for myself) one day:

    How would I solve for [tex]y(x)[/tex] from :shy::

    [tex]\frac{{dy}}{{dy}} = \frac{x}{{f - y + \sqrt {x^2 + \left( {f - y} \right)^2 } }} [/tex]

    where "f" is a constant :smile:
     
  2. jcsd
  3. Nov 5, 2005 #2

    saltydog

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    Hello Bomba. May I suggest you use standard nomenclature: Reserve the 'f' for a function and just use 'k' for a constant:

    [tex]\frac{dy}{dx}=\frac{x}{(k-y)+\sqrt{x^2+(k-y)^2}}[/tex]

    Note that the (k-y) expression occurrs twice. How about using a change of variables for starters:

    [tex]u=k-y[/tex]

    Do that, then note that even though you have a square root, the equation is still homogeneous of degree 1. Such equations are normally solved by letting:

    [tex]u=xv[/tex]

    Make that substitution, do the algebra, simplify, separate variables, integrate, switch back to u, then back to y.:smile:
     
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