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2nd Order Differential Equation (Complex)

  1. Jun 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the general solution to the following differential equation:

    [itex]\frac{d^2 y}{dt^2} - 2 \frac{dy}{dt} + 2y =e^t[/itex]

    The correct answer must be: [itex]y(t) = C_1 e^t \cos t + C_2 e^2 \sin t +e^t[/itex]

    3. The attempt at a solution

    I haven't been able to get the correct answer so far. The eigenvalues are:

    [itex]\lambda^2-2 \lambda +2 = 0[/itex]

    [itex]\lambda = \frac{2 \pm \sqrt{4-8}}{2} = 1 \pm i[/itex]

    So the general solution of the corresponding homogeneous equation is:

    [itex]y_h(t) = C_1 e^{(1-i)t} + C_2 e^{(1-i)t} [/itex]

    And with inspection a particular solution is yp = et, so the general solution must be:

    [itex]y(t) = C_1 e^{(1-i)t} + C_2 e^{(1-i)t} + e^t[/itex]

    Using Euler's formula:

    [itex]y(t) = C_1 \frac{e^t}{\cos t + i \sin t} + C_2 e^t (\cos t + i \sin t)+ e^t[/itex]

    So, why is my answer so different from the correct answer provided? What do I have to do in order to get the correct answer?

    Any help is really appreciated.
     
  2. jcsd
  3. Jun 6, 2012 #2

    sharks

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    The general solution of the reduced equation should be:
    [tex]y_c=e^{t}(C_1 \cos t + C_2 \sin t)[/tex]Then, use the method of variation of parameters to find the particular integral.
    I think you mistyped the answer as i got the general equation: [itex]y(t) = C_1 e^t \cos t + C_2 e^t \sin t +e^t[/itex]
     
    Last edited: Jun 6, 2012
  4. Jun 6, 2012 #3

    LCKurtz

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    The homogeneous solution should be written$$y_h(t) = C_1 e^{(1+i)t} + C_2 e^{(1-i)t} + e^t
    =C_1e^te^{it}+C_2e^te^{-it} =
    C_1e^t(\cos t + i \sin t)+C_2e^t(\cos t - i\sin t)$$from which you can get the sine - cosine form by collecting terms and renaming the constants.
     
  5. Jun 7, 2012 #4
    Yest, that was a typo I meant et not e2.

    Thanks a lot LCKurtz for the hint, I finally got the correct expression.
     
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