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Math Help - Undetermined Coefficients question

  1. #1
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    Undetermined Coefficients question

    y^{\prime \prime} + 81 y = 9 \sin(9 x) + 9 + e^{9 x}

    My Yp expression is:
    <br />
C_{1}x\cos(9 x) + C_{2} x\sin(9 x) + C_{3} + C_{4}e^{9 x}<br />


    But for this one,
    y^{\prime \prime} - 25 y = 5 \cos(5 x) + 5 + e^{5 x}
    its:
    C_{1}\cos(5 x) + C_{2} \sin(5 x) + C_{3} + C_{4}x e^{5 x}

    For the second one, the expression for cos(5x) is written as [tex] C_{1}\cos(5 x) + C_{2} \sin(5 x)

    but for the first one, sin(9x) its :
    <br />
C_{1}x\cos(9 x) + C_{2} x\sin(9 x)<br />

    Why does the second instance have an x infront of the cos and sin? Whats the rule here?
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  2. #2
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    The rule here is that functions that show up in the homogeneous solution (that is, the differential equation LHS = 0), need to be "enhanced" in order not to be annihilated by the LHS's operator when you plug it into the DE. Notice that in your second yp, there's an x multiplying the exponential. That's because in that case, the exponential is already a part of the homogeneous solution. In the first problem, the trig functions are both in the homogeneous solution. Make sense?
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  3. #3
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    Oh Thank you!
    However, when I solve my homo equation for the first one, I get
    ACos(18x)+BSin(18x)
    Which isnt quite whats on the RHS. Does it matter whats inside the cos and sin?
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  4. #4
    A Plied Mathematician
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    It does matter what's inside the trig functions, and I don't think your homogeneous solution is correct. Check it again, if you would, please.
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