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Variable Coefficient PDEs and Continuity of the General Solution

  1. Sep 23, 2012 #1
    Variable Coefficient PDEs

    My homework question:

    "Find the general solution of ##xu_{x} + 4yu_{y} = 0## in ##{(x,y)\neq(0,0})##; when is this solution continuous at (0,0)?"

    ##\frac{dx}{dy} = \frac{x}{4y}##

    ##\frac{dx}{x} = \frac{dy}{4y}##

    Integrating both sides, we find:

    ##lnx + c = \frac{1}{4} lny + d##

    ##lnx + c = lny^{\frac{1}{4}} + d##

    ##lnx + b= lny^{\frac{1}{4}}##, where ##b = c-d##

    ##e^(lnx + b) = e^(lny^{\frac{1}{4}})##

    ##x e^b = y^{\frac{1}{4}}##, but ##e^b## is a constant, so we say ##e^b = C##

    ##Cx = y^{\frac{1}{4}}##

    ##(Cx)^4 = y##


    ##u(x,y) = u(x,[Cx]^4)##
    ##\frac{du(x, [Cx]^4)}{dx} = \frac{du}{dx} + \frac{du}{dy}\frac{dy}{dx} = \frac{du}{dx} + \frac{du}{dy}\frac{4y}{x} = xu_{x} + 4yu_{y} = 0##; this is the initial question, so we know that this works. Now we know:

    ##u(x,y) = u(x, [Cx]^4) = f(C)##, but we can solve for C in terms of x and y to find ##C = \frac{y^(\frac{1}{4})}{x}## so we have the general solution:

    ##u(x,y) = f(\frac{y^(\frac{1}{4})}{x})##

    In terms of continuity is this just asking when the argument of the general solution f is equal to 0? If so, then the general solution would be discontinuous at any point ##(x,y) = (0,y)## or any point along the vertical line described by ##x = 0##. How can I make this function continuous at ##(x,y) = (0,0)##, or better - how can I make it continuous at this point given that f is an arbitrary function; do I use the notion that f is continuous at this point as a boundary condition?

    EDIT

    I think it seems reasonable that I would have to take the limit of f as ##(x,y)## approaches ##(0,0)##. If I find that the limit along two separate paths do not agree then the function is not continuous at this point...(or that just means that the limit does not exist - just because the limit at a point exists does not necessarily mean that the function is continuous, such as at a point discontinuity, correct?)
     
    Last edited: Sep 23, 2012
  2. jcsd
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