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Math Help - Second Order Non-Homogenous Differential Equations

  1. #1
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    Second Order Non-Homogenous Differential Equations

    I have 3 non-homogeneous differential equations. The first i have done myself and believe to be correct, but can someone just confirm. The second and third i just need some guidance on what form the particular solution will take and the rest i will try to do myself:

    1) y" - y' - 12y = e^6x here i used yp = ce^6x and got the general solution as yh = Ae^-3x + Be^4x + e^6x/18

    2) y" - y' -6y = e^-x + 12x and yp = ???? y(0)=1, y'(0)=-2

    3) y" + 2y' + 5y = cos3x and yp = ???? y(0)=1, y'(0)=0
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  2. #2
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    Re: Second Order Non-Homogenous Differential Equations

    2) I would try

    y_{p}=Ae^{-x}+Bx+C.

    3) Try

    y_{p}=A\cos(3x)+B\sin(3x).
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    Re: Second Order Non-Homogenous Differential Equations

    Quote Originally Posted by NFS1 View Post
    I have 3 non-homogeneous differential equations. The first i have done myself and believe to be correct, but can someone just confirm. The second and third i just need some guidance on what form the particular solution will take and the rest i will try to do myself:

    1) y" - y' - 12y = e^6x here i used yp = ce^6x and got the general solution as yh = Ae^-3x + Be^4x + e^6x/18

    2) y" - y' -6y = e^-x + 12x and yp = ???? y(0)=1, y'(0)=-2

    3) y" + 2y' + 5y = cos3x and yp = ???? y(0)=1, y'(0)=0
    In 1) you seem forgotten to add a constant. ({e^6}/{144}).


    Give us your solutions for all the three, and we will try to help more.
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    Re: Second Order Non-Homogenous Differential Equations

    surely is the constant not e^6x/18 as

    >>yp=ce^6x so y'p=6e^6x and y"p=36e^6x
    >>sub into the diff. equ. to get (36c - 6c - 12c)e^6x=e^6x
    >> so 18c=1 hence c=1/18 therefore yh = Ae^-3x + Be^4x + e^6x/18

    so how is it e^6x/144??
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    Re: Second Order Non-Homogenous Differential Equations

    Your solution to the first problem is correct, I believe, though I would encourage you to use parentheses around the arguments of functions. That is, write it as y = Ae^-3x + Be^4x + e^(6x)/18, or better yet use LaTeX.
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    Re: Second Order Non-Homogenous Differential Equations

    Quote Originally Posted by Ackbeet View Post
    2) I would try

    y_{p}=Ae^{-x}+Bx+C.

    3) Try

    y_{p}=A\cos(3x)+B\sin(3x).
    2) so >> yp=Ae^-x +Bx + C
    >>yp = ce^-x + 12Bx + c ???
    >> y'p = -ce^-x + 12B and y"p= ce^-x

    is the above correct???
    Last edited by NFS1; August 11th 2011 at 04:09 PM.
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    MHF Contributor Siron's Avatar
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    Re: Second Order Non-Homogenous Differential Equations

    You can take a look @ wolphram alpha if you want.
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    Re: Second Order Non-Homogenous Differential Equations

    Quote Originally Posted by NFS1 View Post
    2) so >> yp=Ae^-x +Bx + C
    >>yp = ce^-x + 12Bx + c ???
    >> y'p = -ce^-x + 12B and y"p= ce^-x

    is the above correct???
    Not sure why you have the 12 in there, but your working is on the right track to getting the correct answer.
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    Re: Second Order Non-Homogenous Differential Equations

    Quote Originally Posted by Ackbeet View Post
    Not sure why you have the 12 in there, but your working is on the right track to getting the correct answer.
    I thought as the rhs of the 2nd euation is e^-x + 12x so yp= ce^-x + 12b + c
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    Re: Second Order Non-Homogenous Differential Equations

    Quote Originally Posted by NFS1 View Post
    I thought as the rhs of the 2nd euation is e^-x + 12x so yp= ce^-x + 12b + c
    Perhaps, but it's going to be absorbed into the b anyway. Incidentally, mathematics is case-sensitive. B is not equal to b, so I would advise paying more attention to having a consistent case for variables throughout your work.
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  11. #11
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    Re: Second Order Non-Homogenous Differential Equations

    ok so im having difficulty on solving the 2nd, below is what i have done:

    diff equa. >> y" - y' -6y = e^-x + 12x
    particular equa.>> yp = Ae^-x + Bx + C so y'p= -Ae^-x + B and y"p= Ae^-x
    sub into diff eq >> Ae^-x -(-Ae^-x + B) - 6(Ae^-x + Bx + C) = e^-x + 12x
    simplify >> -4e^-x - 6Bx - 6C - B = e^-x + 12x

    how would i solve from here??
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  12. #12
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    Re: Second Order Non-Homogenous Differential Equations

    Quote Originally Posted by NFS1 View Post
    ok so im having difficulty on solving the 2nd, below is what i have done:

    diff equa. >> y" - y' -6y = e^-x + 12x
    particular equa.>> yp = Ae^-x + Bx + C so y'p= -Ae^-x + B and y"p= Ae^-x
    sub into diff eq >> Ae^-x -(-Ae^-x + B) - 6(Ae^-x + Bx + C) = e^-x + 12x
    simplify >> -4Ae^-x - 6Bx - 6C - B = e^-x + 12x

    how would i solve from here??
    It should be: -4Ae^-x - 6Bx - 6C - B = e^-x + 12x

    Get rid of stuff on the right side.

    (-4A-1)e^-x +(-6B - 12)x + (-6C - B) = 0

    So, (-4A-1) = 0, (-6B - 12) = 0, (-6C - B) = 0 .

    Solve for A, B, C.
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  13. #13
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    Re: Second Order Non-Homogenous Differential Equations

    Quote Originally Posted by SammyS View Post
    It should be: -4Ae^-x - 6Bx - 6C - B = e^-x + 12x

    Get rid of stuff on the right side.

    (-4A-1)e^-x +(-6B - 12)x + (-6C - B) = 0

    So, (-4A-1) = 0, (-6B - 12) = 0, (-6C - B) = 0 .

    Solve for A, B, C.
    ok i have A=-1/4 B==-2 C=-1/3
    subbed into to get general solution as >> yh= Ae^3x + Be^-2x - 1/4e^-x - 2x - 1/3

    subbed in initial conditions to get A= 19/12 and B=0
    so initial value problem is yh = 19/12e^3x + 0e^-2x - 1/4e^-x - 2x - 1/3
    simplified >>yh = 19/12e^3x- 1/4e^-x - 2x - 1/3

    can sum1 confirm if that is correct??
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