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Homework Help: Someone please help me understand this.

  1. Mar 15, 2006 #1
    [tex] \frac{\partial^2 \phi}{\partial x^2} + \lambda \phi = 0 [/tex]

    We have to analyze eigenvalues.

    My prof. gave me this answer for [tex]\lambda>0[/tex] it is
    [tex]c*sin(\sqrt(\lambda))x+d*cos(\sqrt(\lambda))x[/tex]

    Shouldn't it be -[tex]\lambda[/tex] inside the squareroot? If not can someone explain how he got this?
     
    Last edited: Mar 15, 2006
  2. jcsd
  3. Mar 15, 2006 #2
    The problem you'er presenting is homogenic.
    sin and cos are natural solutions for this kind of equations.
    pay attention that if you use diff on sin twice, you get -sin.
    same goes for cos.
    c,d = constants
    you need 2 inputs to get the private solution. otherwisw it's a general solution.
     
  4. Mar 15, 2006 #3
    I know its a general solution but I don't understand how he got the positive [tex]\lambda[/tex] inside the squareroot.
     
  5. Mar 15, 2006 #4
    Looks right to me.. for lambda > 0, the eigenvalue is imaginary.

    Yes, it is [tex]\sqrt(\lambda)[/tex] or [tex]sqrt(/lambda))i[/tex]

    should be square root of neg lambda in the first.. give me a while to get latex right..
     
    Last edited: Mar 15, 2006
  6. Mar 15, 2006 #5

    CarlB

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    Homework Helper

    Your prof was right.

    If you put a minus sign in the square root, you would have to change the sin and cosine to exponential functions. I.e. [tex]C \exp(\sqrt{-\lambda}x)+D\exp(\sqrt{-\lambda}x).[/tex]

    Carl
     
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