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DiffEQ - Linearly Dependent

  1. Feb 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Determine whether the pairs of functions are linear dependent or linearly independent.

    f(x) = x^3
    g(x) = x2|x|

    2. Relevant equations



    3. The attempt at a solution

    g(x)=x^2|x| = x^2*sqrt[x^2] = sqrt[x^6] = x^3

    f'=3x^2
    g'=3x^2

    fg'-f'g = 0

    Linearly Dependent according to me, Linearly independent according to the book.

    I assume it has to do with the absolute value, could someone enlighten me?
     
  2. jcsd
  3. Feb 17, 2010 #2
    Should g'(x) be 2x|x|+x^2(|x|/x) ?
     
  4. Feb 17, 2010 #3
    note that f is positive for some values of x and negative for others whereas g is always positive. if these two were linearly dependent one would be a constant multiple of the other FOR ALL VALUES OF x. can we multiply a fully positive function by a number so that part of it becomes negative? no. therefore no linear dependence. note also this depends on the set over which g and f are defined.
     
  5. Feb 17, 2010 #4
    why are you take the derivative?
     
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