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Linearly Dependent Differential Equations

  1. Mar 18, 2006 #1
    Show directly that the following functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the funtions that vanishes indetically.
    f(x)=17, g(x)= 2sin^2 x, h(x)= 3cos^2 x

    Do you just take the 1st and 2nd derivatives and do the determinate?? I am so confused on how to do this problem. Thanks
     
  2. jcsd
  3. Mar 18, 2006 #2

    0rthodontist

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    You have to find a nontrivial linear combination of the functions that vanishes identically. In other words, you have to find constants A, B, C, not all zero, so that
    A*f(x) + B*g(x) +C*h(x) = 0
    Use trig.
     
  4. Mar 18, 2006 #3
    So basically something like: 3g(x) + 2h(x) =6sin²x + 6cos²x = 6, but then where do I go from there?
     
  5. Mar 18, 2006 #4

    0rthodontist

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    Then you use f(x) to make it 0.
     
  6. Mar 18, 2006 #5
    ok, I got that A17 has to equal -6 to make it zero, so then that would mean A=-6/17, so the final answer would be (-6/17)17 + 3(2sin²x)+2(3cos²x)=0, Correct? And thanks for the help
     
  7. Mar 19, 2006 #6

    0rthodontist

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    Yes, correct.
     
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