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What did I do Wrong?

  1. Mar 2, 2007 #1
    What did I do Wrong??

    1. The problem statement, all variables and given/known data

    Find a particular solution yp of the differential equation
    [tex] 9y''+5y'+2y=sin^2(x) [/tex]

    using the Method of Undetermined Coefficients. Primes denote derivatives with respect to x.

    2. Relevant equations

    [tex] sin^2(x) = \frac{1-cos(2x)}{2} [/tex]

    3. The attempt at a solution

    Possible derivatives
    • A
    • B cos(2x)
    • C sin(2x)

    [tex] y_p = A + Bcos(2x) + Csin(2x) [/tex]

    [tex] y_p' = -2Bsin(2x) + 2Ccos(2x) [/tex]

    [tex] y_p'' = -4Bcos(2x) + -4Csin(2x) [/tex]

    Sub back into original modified with cos substitution on the right

    [tex] 9y''+5y'+2y = \frac{1-cos(2x)}{2} [/tex]

    Becomes

    [tex] 9[-4Bcos(2x) + -4Csin(2x)] + 5[-2Bsin(2x) + 2Ccos(2x)] + 2[A + Bcos(2x) + Csin(2x)] = \frac{1-cos(2x)}{2} [/tex]

    From that I get
    2A = 1/2 => A=1/4

    -36B + 10C + 2B = -1/2
    -36C + -10B + 2C = 0

    Solving for B and C
    Leads to

    [tex]y_p = (1/4)+(17/1056)*cos(2*x)+(5/1056)*sin(2*x) [/tex]

    Where
    B = 17/1056
    and
    C = 5/1056
     
    Last edited: Mar 2, 2007
  2. jcsd
  3. Mar 2, 2007 #2

    arildno

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    Dearly Missed

    Spare us the decimals. Please!!
     
  4. Mar 2, 2007 #3
    Did you check your answer by plugging it into the diff. eq.? (I'm sure you will see that A is wrong at least.)
     
  5. Mar 2, 2007 #4
    I realized I type it wrong when i had A defined, but i had it correct or so I thought in the equation A=1/4
     
  6. Mar 2, 2007 #5

    AKG

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    You solved for A correctly, but B and C are wrong.
     
  7. Mar 2, 2007 #6
    -36B + 10C + 2B = -1/2
    -36C + -10B + 2C = 0

    -34B + 10 C = -1/2
    -10B + -34C = 0

    34B + 115.6C = 0
    125.6C=-1/2

    C = -5/1256

    Therefore B equal

    -10B + -34C = 0

    B = 17/1256


    I am down to one final attempt at this problem can someone confirm this correct?


    [tex]y_p = (1/4)+(17/1256)*cos(2*x)+(-5/1256)*sin(2*x) [/tex]
     
  8. Mar 2, 2007 #7

    Dick

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    That's what I get.
     
  9. Mar 2, 2007 #8
    Algerbra FTW

    ... eh wow I feel stupid

    it worked!
     
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