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Differential Equation NonHomogeneous EQ

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Find a particular solution of the given differential equation.
    y'' + 4y = (sin^2)(x)

    Answer from book: yp(x)= (1/8)(1-xsin2x)


    3. The attempt at a solution

    y'' + 4y = 4(sinx)^2.............(1)

    4(sinx)^2=2 - 2cos2x

    Homogeneous solution:
    r^2+4=0
    r^2= -4
    r=+-2i ==>
    yh=C1cos(2x)+C2sin(2x)

    Particular solution:
    yp=A+Bxcos2x+Cxsin2x
    yp' =Bcos2x-2Bxsin2x+Csin2x+2Cxcos2x

    yp" = -2Bsin2x -2Bsin2x-4Bxcos2x
    +2Ccos2x+2Ccos2x - 4Cxsin2x

    Substitute y and y" in (1) ==>
    4A - 4Bsin2x+4Ccos2x=2 - 2cos2x ==>

    A=1/2
    B=0
    C= -1/2

    yp=1/2 - (1/2)xsin2x

    This is answer is different from the book. What am I doing wrong?
     
  2. jcsd
  3. Mar 8, 2010 #2

    Mark44

    Staff: Mentor

    How did the right side go from sin2(x) to 4sin2(x)?

    What you need to do is rewrite the right side using a double angle identity, sin2(x) = 1/2 - (1/2)cos(2x).
    Your work below suggests that you have done this, but I don't see that you mention it anywhere.
    This is fine.
    This (above) is what you need.
    The only mistake I see is that you changed the right side of the DE from sin2(x) to 4sin2(x). That could be causing problems for you in solving for the coefficients A, B, and C.
     
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