Equation w/ Homogeneous Coefficients - y=ux substitution

  1. Differential Equation w/ Homogeneous Coefficients - y=ux substitution

    I am teaching myself, this problem is from ODEs by Tenenbaum and Pollard. This is not homework for a class.

    1. The problem statement, all variables and given/known data

    (x+y){dx} - (x-y){dy} = 0

    2. Relevant equations

    y=ux, {dy} = u{dx} + x{du}

    3. The attempt at a solution

    Substitution should lead to a separable equation in x and u:

    (x+ux){dx} + (ux - x)(u{dx} + x{du}) = 0;
    x(u+1)dx + u2x{dx} + ux2{du} - ux{dx} - x2{du} = 0;
    xu(2){dx} + x2(u - 1){du} = 0;
    -1/x{dx} = (u - 1)/(u2 + 1){du}

    Okay, I am assuming that I am correct up to this point but the answer given by the text is:

    Arc tan(y/x) - 1/2log(x2 + y2) = c

    I understand where the Arc tan(y/x) comes from - the (1/(u2 + 1)). I am having trouble with the 1/2 log(x2 + y2) - where does the -log(x) go?

    I have a couple of other problems in the same form as this which arise in isogonal trajectories and I don't want to just skip over this because I am obviously having a problem with these integrations.
    Last edited: Nov 12, 2008
  2. jcsd
  3. HallsofIvy

    HallsofIvy 41,264
    Staff Emeritus
    Science Advisor

    Re: Differential Equation w/ Homogeneous Coefficients - y=ux substitution

    so -1/x dx= u/(u2+ 1)du - 1/(u2+1 du
    The left side give -ln(x), of course. Then anti-derivative of -1/(u2+ 1) is -arctan(u) but to integrate u/(u2+ 1) let v= u2+ 1, dv= 2udu and the integral becomes udu/(u2+ 1)= dv/(2v)= (1/2)ln(v)= (1/2) ln(u2+ 1)= (1/2)ln((y2/x2+ 1)= (1/2)ln((x2+ y2)/x2= (1/2)ln(x2+ y2)- ln(x).

  4. Yes, thank you - I was not splitting up the ration into two equations and then integrating. I will try to be more observant from now on.
  5. tiny-tim

    tiny-tim 26,016
    Science Advisor
    Homework Helper

    Hi ZachN! :smile:

    As an alternative method … always look at the answer … it may give you a clue as to an easy substitution …

    in this case, the answer uses tan-1y/x and x2 + y2, so the obvious substitution is into polar coordinates, r and θ.

    Try it and see. :smile:
  6. I'll try polar coordinates.
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