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Help with solving system of DE's

  1. May 13, 2013 #1
    My DE skills are a bit rusty, and I need some help remembering how to handle a system such as:

    [itex]\dot{x_1}=x_2[/itex]
    [itex]\dot{x_2}=-2x_1-3x_2+sint+e^t[/itex]

    I have found the homogeneous solution to be (sorry I don't know how to do matrices here):

    [itex]c_1\left\{e^{-t}\right\}+c_2\left\{e^{-2t}\right\}[/itex]
    [itex]c_1\left\{-e^{-t}\right\}+c_2\left\{-2e^{-2t}\right\}[/itex]

    From what I've found online I should guess a particular solution form:

    [itex]x_{p}=Asin(t)+Bcos(t)+Ce^{t}[/itex]

    Where A, B, and C are 2x1 matrices of constants [itex]a_1, a_2, b_1, b_2, c_1, c_2[/itex]

    Is this correct?
    Then rewrite the original in the form:

    [itex]\dot{x_{p}}=Ax_{p}+g[/itex]

    Then differentiate the guess and substitute back into the above.

    Assuming this is all correct, what are the next steps in finding the general solution?
     
  2. jcsd
  3. May 19, 2013 #2
    Assuming that your system can be written as the matrix form [itex]\dot{X}=AX+F(t).[/itex] , e.g. X=[x1 x2]t etc
    Then the general solution for this equation should be (if I'm not mistaken)
    [tex]X(t)=e^{At}C+e^{At}\int_0^t e^{-As}F(s) ds [/tex]

    Your particular solution in this case is then
    [tex]X_p(t)=e^{At}\int_0^t e^{-As}F(s) ds [/tex]
     
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