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Math Help - Linear Differential Equations

  1. #1
    Junior Member
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    Mar 2008
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    Linear Differential Equations

    Hello, I'd appreciate some help finding the general solution of the differential equation:

    \frac{d^2 y}{dx^2} + 4y = \sin{x}

    a^2 + 4a =0
    (a + 2)^2 = 4
    a = 0, a = 4

    y = Ce^4x + B

    But how do I find the particular integral?

    Edit: Sorry it's been a while since I've Latex'd, and I'm kinda in a hurry, so I'll leave it as it is.
    Last edited by Nyoxis; April 26th 2009 at 12:40 PM.
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  2. #2
    MHF Contributor
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    Quote Originally Posted by Nyoxis View Post
    Hello, I'd appreciate some help finding the general solution of the differential equation:

    \frac{d^2 y}{dx^2} + 4y = \sin{x}

    a^2 + 4a =0
    (a + 2)^2 = 4
    a = 0, a = 4

    y = Ce^4x + B

    But how do I find the particular integral?

    Edit: Sorry it's been a while since I've Latex'd, and I'm kinda in a hurry, so I'll leave it as it is.
    Hi

    First the characteristic equation is \lambda^2 + 4 = 0 which leads to \lambda = -2i

    y = A \cos 2x + B \sin 2x

    The particular solution can be found as C \sin x

    -C \sin x + 4C \sin x = \sin x leads to C = \frac13
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