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Solve by Method of Characteristics

  1. Oct 5, 2011 #1

    syj

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    1. The problem statement, all variables and given/known data
    [itex]u\frac{\delta u}{\delta x}[/itex]+[itex]\frac{\delta u}{\delta y}[/itex] =1

    [itex]u|_{x=y}=\frac{x}{2}[/itex]

    2. Relevant equations

    the characteristic equations to solve are:

    [itex]\frac{dx}{ds}=u[/itex]

    [itex]\frac{dy}{ds}=1[/itex]

    [itex]\frac{du}{ds}=1[/itex]

    3. The attempt at a solution

    I got the following equations from the characteristic equations:

    [itex] \frac{dx}{ds}=u[/itex] means [itex]\frac{d^2x}{ds^2}=\frac{du}{ds}=1[/itex]
    after integrating twice i get [itex]x=\frac{1}{2}s^2+ks+x_0[/itex]

    [itex] y = s +y_{0} [/itex]

    [itex] u = s +u_{0} [/itex]

    and here is where I am now stuck.
    I also dont know how to apply the condition given : [itex]u|_{x=y}=\frac{x}{2}[/itex]
     
    Last edited: Oct 5, 2011
  2. jcsd
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