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Solution of a nonhomogeneous equation

  1. Jun 19, 2011 #1
    (i d0nt kn0w how to use LaTeX)

    1)D^2(D-1)y=3e^x+sinx

    for yc:
    let y=e^mx
    D^2(D-1)e^mx=0
    m^2(m-1)e^mx=0
    f(m)=0
    m^2(m-1)=0
    m=0,0,1

    yc= C1+C2x+C3e^x

    for yp:
    R(x)=3e^x+sinx
    m'=1,+/- 1i

    yp=Axe^x+Bcosx+Csinx
    yp'=A(xe^x+e^x)-Bsinx+Ccosx
    yp"=A(xe^x+2e^x)-Bcosx-Csinx

    D^2(D-1)yp=3e^x+sinx

    Guys can you help me find the value of A, B and C. I had a hard time simplifying the terms and can't equate the coefficients. Thanks for analyzing my post this far. :D
     
  2. jcsd
  3. Jun 19, 2011 #2

    vela

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    You're doing fine so far. You need to differentiate one more time because

    D2(D-1)yp = (D3-D2)yp=yp'''-yp''
     
    Last edited: Jun 19, 2011
  4. Jun 19, 2011 #3
    Thanks for your help vela! I simply overlooked the problem and didn't get the third derivative. :D
     
  5. Jun 19, 2011 #4

    HallsofIvy

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    Here's how I would have done it:
    If [itex]D^@(D-1)y= 3e^x+ sin(x)[/itex] then [itex]D(D-1)y= 3e^x- cos(x)+ C[/itex] and [itex](D-1)y= 3e^x- sin(x)+ Cx+ D[/itex]

    Then solve the characteristic equation m- 1= 0 to get m= 1 which means that [itex]e^x[/itex] is a solution to the associated homogeneous equation. Since part of the right side is [itex]e^x[/itex] look for a solution of the form [itex]Axe^x+ Bsin(x)+ Ccos(x)+ Ex+ F[/itex].
     
  6. Jun 19, 2011 #5
    Can you figure out my mistake. The answer is supposed to be:
    y=C1e^-x+C2e^-2x+6x^2-18x+21

    Here is the problem and my solution:

    2) (D^2+3D+2)y=12x^2

    for yc
    let y=e^mx
    (D^2+3D+2)e^mx=0
    (m^2+3m+2)e^mx=0
    f(m)=0
    m^2+3m+2=0
    m=-1,-2

    yc=C1e^-x+C2e^-2x

    for yp
    R(x)=12x^2
    m'=0,0,0
    yp=A+Bx+Cx^2
    yp'=B+2Cx^2
    yp"=4Cx^2

    (D^2+3D+2)yp=12x^2

    4Cx^2+3B+6Cx^2+2A+2Bx+2Cx^2=12x^2

    3B+12Cx^2+2A+2Bx=12x^2

    Equating coefficients
    i get C=1, B&A=0

    Giving:
    yp=x^2

    Answer
    y=C1e^-x+C2e^-2x+x^2
    (which is wr0ng)

    Thanks for any help!
     
  7. Jun 19, 2011 #6
    P.s. An0ther problem was posted.
     
  8. Jun 19, 2011 #7

    vela

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    You didn't calculate the derivatives of yp correctly.
     
  9. Jun 20, 2011 #8
    Alright, i get it. :D I'm used to differentiate e^n terms and overlooked x^n terms. I'm becoming reckless. Thanks vela.
     
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