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Math Help - Undetermined coefficient problem

  1. #1
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    Undetermined coefficient problem

    Find a particular solution of y^{(3)}+y'=2-sinx

    Solution:

    The homogeneous solution is y_h(x)=C_1+C_2 \cos x + C_3 \sin x

    Now, my first guess for the particular solution is: y_p(x)=A+B \sin x+C \cos x

    I have all terms repeated here, so multiply by x to all, obtains:
    y_p(x)=Ax+B x \sin x + C x \cos x

    Would that be the right guess? Somehow the answer that I obtain from this doesn't match with the one in the book.

    Thanks.
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  2. #2
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    Helo, tttcomrader!

    You're doing great!


    Find a particular solution of: . y'''+y'\:=\:2-\sin x

    Solution:

    The homogeneous solution is: . y_h(x)\:=\:C_1+C_2 \cos x + C_3 \sin x . . . . Right!

    Now, my first guess for the particular solution is: . y_p(x)\:=\:A+B \sin x+C \cos x

    I have all terms repeated here, so multiply by x to all, obtains:
    . . y_p(x)\:=\:Ax + Bx\sin x + Cx\cos x

    Would that be the right guess? . . . . Yes!
    You're probably making simple errors here and there.
    It took me a few tries to get it right . . .

    I finally got: . A = 2,\;B = \tfrac{1}{2},\;C = 0 \quad\Rightarrow\quad\boxed{ y_p \:=\:2x + \tfrac{1}{2}x\sin x}

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