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First-order inhomogeneous PDE

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Assume ut+cux = xt, u(x,0) = f(x) for t>0. Find a formula for u(x,t) in terms of f, x, t, and c.

    3. The attempt at a solution
    I don't really follow what the professor is doing in class, and his office hours and the textbook weren't much more help, so the only thing I know about PDE's is what I've read online. That said:

    [tex]\frac{du}{dr}[/tex] = [tex]\frac{dx}{dr}[/tex]ux+[tex]\frac{dt}{dr}[/tex]ut

    [tex]\frac{dt}{dr}[/tex]=1
    t=r
    [tex]\frac{dx}{dt}[/tex]=c
    x=ct+c'
    x0=c'
    x=ct+x0

    [tex]\frac{du}{dr}[/tex]=xt=ct2+x0t
    [tex]\int[/tex]du=[tex]\int[/tex](ct2+x0t)dr
    u(x,t) = ct2r+x0tr+c''
    u(x0,0) = 0+0+c'' = f(x0)
    u(x,t) = ct2r+x0tr+f(x0)
    = ct3+(x-ct)t2+f(x-ct)
    = xt2+f(x-ct)

    but when I calculate ut and ux and substitute into the original equation I do not get xt.

    Any pointers would be much appreciated!
     
  2. jcsd
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