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Math Help - Second Order differential equation problem

  1. #1
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    Second Order differential equation problem

    y''+9y=30sin(3x)

    I'm told to use the method of solving y''+9y=30e^(3ix), but I'm not sure what to do after I find (D-3)(D-3)y=30e^(3ix).

    Could I have some help?
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  2. #2
    MHF Contributor chisigma's Avatar
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    In terms of Laplace Transform the particular integral of...

     y^{''} + 9\cdot y = 30 \cdot \sin 3x (1)

    ... is written as...

    Y_{p} (s) = \frac{10}{(s^{2}+9)^{2}} (2)

    ... so that is...

    y_{p} (x) = \mathcal {L} \{Y_{p} (s)\} = \frac {5}{27}\cdot (\sin 3x - 3x\cdot \cos 3x) (3)

    Alternatively [and is is a not confortable job ...] You can search for function of the type...

    y_{p} (x) = x \cdot (\alpha\cdot \cos 3x + \beta\cdot \sin 3x) (4)

    ... that satisfies (1)...

    Kind regards

    \chi \sigma
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