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Simple one on ODE

  1. Jan 4, 2006 #1
    Hi,

    I have

    [tex]
    y'' - y' = \frac{1}{e^{x} + 1}
    [/tex]

    What I did:

    [tex]
    \lambda^2 - \lambda = 0
    [/tex]

    [tex]
    \lambda_1 = 0
    [/tex]

    [tex]
    \lambda_2 = 1
    [/tex]


    FSS = [itex][1, e^x][/itex]

    [tex]
    y = a + be^x
    [/tex]

    [tex]
    y' = a' + b'e^x + be^x
    [/tex]

    Putting first two terms zero, I get

    [tex]
    y'' = b'e^x + be^x
    [/tex]

    [tex]
    y'' - y' = b'e^x + be^x - be^x = b'e^x = \frac{1}{e^x + 1}
    [/tex]

    [tex]
    b' = \frac{1}{e^{2x} + e^x}
    [/tex]

    From the second condition

    [tex]
    a' + b'e^x = 0
    [/tex]

    I get

    [tex]
    a' = -\frac{e^x}{e^{2x} + e^x} = 1 - \frac{e^x}{1+e^x}
    [/tex]

    Anyway,

    [tex]
    a' + b'e^x = 0
    [/tex]

    now isn't true. Where do I do the mistake?

    Thank you very much.
     
  2. jcsd
  3. Jan 4, 2006 #2

    hotvette

    User Avatar
    Homework Helper

    Your mistake is in simplifying a'. Check it again.
     
  4. Jan 4, 2006 #3
    Thank you hotvette! I got it.
     
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