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Use the differential operator to solve this differential equation

  1. Mar 5, 2012 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Use the differential operator method to find a fundamental set of solutions {y_1(x), y_2(x)} of the equation

    d^2 y/dx^2 - 18 dy/dx + 90y = 0.


    2. Relevant equations
    Differential operator method.


    3. The attempt at a solution
    I have a huge problem understanding how to use the differential operator method. I can successfully complete this problem using the characteristic equation:

    r^2 - 18r + 90 = 0
    r = 9 +/- 3i
    y_1(x) = e^(9x) * cos(3x)
    y_2(x) = e^(9x) * sin(3x)

    but I really need to understand how to use the differential operator and I don't get anything I read on the internet or in my text book.

    A comparison (with contrasting) to the method with the characteristic equation would be GREATLY appreciated!
    Thanks in advance!
     
    Last edited: Mar 5, 2012
  2. jcsd
  3. Mar 5, 2012 #2

    jambaugh

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    Science Advisor
    Gold Member

    Since the equation is constant coefficient, the characteristic polynomial is also the differential operator form:

    Your equation takes the form:
    [itex] [D^2 -18D +90]y = 0[/itex], where [itex] D = \frac{d}{dx}[/itex]
    Factoring the characteristic polynomial becomes a factorization of the differential operator:
    [itex] (D-r_1)(D-r_2)y = 0[/itex]
    you can then successively integrate:
    Solve the first order equation: [itex](D-r_1)y=0[/itex] and let [itex]y_1[/itex] be your general solution.
    Then solve the first order equation [itex](D-r_2)y = y_1[/itex] and let that solution be [itex] y_2[/itex]

    Then observer that [itex](D-r_1)\left\{ (D-r_2) y_2 \right\} = (D-r_1) {y_1} = 0[/itex] so [itex]y_2[/itex] is a solution to your original equation. If you gave the most general solution in each step you have the most general solution to the original problem.
     
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