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1st order PDE

  1. Apr 3, 2014 #1
    1. The problem statement, all variables and given/known data
    suppose [tex]u(x,y)[/tex] satisfies the partial differential equation:
    [tex]-4y\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0[/tex]
    Find the characteristic curves for this equation and name the shape they form



    3. The attempt at a solution
    [tex]\frac{dy}{dt}=1 \Longrightarrow y=t+y_0[/tex]
    [tex]\frac{dx}{dt}=-4y=-4(t+y_0) \Longrightarrow x=-2t^2-4y_0t+x_0[/tex]

    What can I say about the shape these curves form?
     
  2. jcsd
  3. Apr 3, 2014 #2

    pasmith

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    Homework Helper

    [itex]y[/itex] is a first-order polynomial in [itex]t[/itex]. [itex]x[/itex] is a quadratic in [itex]t[/itex]. Therefore [itex]x[/itex] is a quadratic in [itex]y[/itex]. What shape does the graph of a quadratic function have?
     
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