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Finding if a solution exists

  1. May 24, 2004 #1
    hi
    this is my first attempt at using the latex commands ..
    consider this Cauchy Problem:
    [tex]
    u_x\exp y +u_y\exp(x)=1
    [/tex]
    [tex]
    u(t,tk^2)=k\exp(-t)
    [/tex]
    where k is a constant.
    forgive me i have given up using latex as i am too slow, it would take me a week to type it up at this rate. anyway
    i am trying to find the values of k where
    a) this problem has a unique solution
    b) no solutions

    for part a) i know that it all boils down to showing where the vector (1,k^2) is never parallel to ( exp(t*k^2), exp(t) )
    i eventually get to:
    t cant be equal to 2ln(k)/(1-k^2), however the question asks for values of k and i havent done that. but i dont know any way to find explicit values of k

    for part b) i just said t=2ln(k)/(1-k^2)

    is it reasonable to give the answer in the form i have? i'm not sure
    hope you can help

    thanks,
    vladimir
     
  2. jcsd
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