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ODE ( 2nd order nonhomogeneous equation)

  1. Feb 24, 2009 #1
    1. The problem statement, all variables and given/known data
    By using the method of undetermined coefficients,find the particular solution of
    y''+y'+y=(sin x)^2


    2. Relevant equations
    i know how to determine the particular solution IF it is sin x.
    Ex: sin x ====> Asin x + B cos x (particular)

    but i wonder how to determine the (sin x)^2


    3. The attempt at a solution
     
  2. jcsd
  3. Feb 24, 2009 #2
    i haven't actually seen a problem like this come up, but i think similarly to finding the particular solution to something like lhs=t^2 is A+Bt+Ct^2, it'd be something like:

    Asin(x)+Bcos(x)+Ccos^2(x)+Dsin^2(x)

    i'm not 100% sure, but i'd try something like that and see if it works out. good luck!
     
  4. Feb 24, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You can't assume a solution of the form cos2 x or sin2 x because sine or cosine squared are not of the form that gets as solutions to a homogeneous equation with constant coefficients. However, you CAN use a trigonometric identity. Since cos(a+ b)= cos(a)cos(b)- sin(a)sin(b), taking a= b= x, cos(2x)= cos2(x)- sin2(x). Replacing cos2(x) by 1- sin2(x), cos(2x)= 1- 2sin2(x) so sin2(x)= (1/2)(1- cos(2x)). Look for a particular solution of the form A+ Bsin(2x)+ Ccos(2x).
     
  5. Mar 3, 2009 #4
    Thx for helping ! i did solve the Q:!!)
     
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