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    Say in a linear 2nd DE y"+y'+y = f(x)
    where f(x) = 4sin(3x)
    The complementary function is Asin(4x)+Bcos(4x)
    would i use a particular integral of asin(3x)+bcos(3x)?

    What if complementary function was Asin(3x)+Bcos(3x)
    Would i then use a PI of axcos(3x)?
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    YES to your 4th line...
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    (Original post by Zenarthra)
    Say in a linear 2nd DE y"+y'+y = f(x)
    where f(x) = 4sin(3x)
    The complementary function is Asin(4x)+Bcos(4x)
    would i use a particular integral of asin(3x)+bcos(3x)?

    What if complementary function was Asin(3x)+Bcos(3x)
    Would i then use a PI of axcos(3x)?
    First part: yes

    Second part: it ought to be axsin3x + bxcos3x
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    if by complimentary you mean the solution to the homogenuous part, then you`d have to choose the form of your last line, but a polynomial of degree one, i,e,:

    x*(A \sin(3x)+B \cos(3x))


    corrected - big up tinyhobbit
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    You need to multiply by the independent variable if (part of) the particular integral appears in the complementary function. additionally if this is true and the solutions to the characteristic equation are repeated roots then multiply by the independent variable squared.
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    (Original post by tiny hobbit)
    First part: yes

    Second part: it ought to be axsin3x + bxcos3x
    (Original post by Hasufel)
    YES to your 4th line...
    (Original post by poorform)
    You need to multiply by the independent variable if (part of) the particular integral appears in the complementary function. additionally if this is true and the solutions to the characteristic equation are repeated roots then multiply by the independent variable squared.



    wHY DOES MY BOOK SAY THIS?
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    (Original post by Zenarthra)


    wHY DOES MY BOOK SAY THIS?
    What exactly is your issue with 'this'
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    (Original post by Phichi)
    What exactly is your issue with 'this'
    I dont have an issue but is it correct?
    Why have some members told me to use a PI of axsin(3x)+bx(sin3x)?
    when f(x)=4sin(3x) is contained in the cf?
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    Yeah this is confusing me too as I've never seen a question on it. Judging by the textbook, if the Complementary Function is acosx + bsinx and the right hand side is for example 2cosx then the PI is axsinx, and if the right hand side is 2sinx then the PI is axcosx.

    Really hoping they don't give us a question on this tomorrow as I have never used it before!
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    (Original post by Zenarthra)
    I dont have an issue but is it correct?
    Why have some members told me to use a PI of axsin(3x)+bx(sin3x)?
    when f(x)=4sin(3x) is contained in the cf?
    (Original post by FeelsToWaltz)
    Yeah this is confusing me too as I've never seen a question on it. Judging by the textbook, if the Complementary Function is acosx + bsinx and the right hand side is for example 2cosx then the PI is axsinx, and if the right hand side is 2sinx then the PI is axcosx.

    Really hoping they don't give us a question on this tomorrow as I have never used it before!
    If f(x) takes the form Asin(\lambda x) or Bsin(\lambda x) use the P.I Psin(\lambda x) + Qcos(\lambda x)

    But if you have either C_1sin(\lambda x) or C_2cos(\lambda x) already in the CF use Pxsin(\lambda x) + Qxcos(\lambda x)
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    (Original post by Zenarthra)
    I dont have an issue but is it correct?
    Why have some members told me to use a PI of axsin(3x)+bx(sin3x)?
    when f(x)=4sin(3x) is contained in the cf?
    I'd go with that PI myself.

    (This is really something you should be able to test out for yourself! Make up a DE with the type of f(x) where you have an issue; try the different possible PIs that you are debating, and see what comes out. You should be able to see from the end result which type of PI is going to work and which one is going to give you problems because you either end up with 0 = f(x) or something like acos(mx) + bxsin(mx) = Asin(mx) which can't be solved.)
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    (Original post by Phichi)
    If f(x) takes the form Asin(\lambda x) or Bsin(\lambda x) use the P.I Psin(\lambda x) + Qcos(\lambda x)

    But if you have either C_1sin(\lambda x) or C_2cos(\lambda x) already in the CF use Pxsin(\lambda x) + Qxcos(\lambda x)
    If Csin(Ax) is contained in cf why shouldnt we use Dxcos(Ax)?
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    (Original post by Zenarthra)


    wHY DOES MY BOOK SAY THIS?

    (Original post by Phichi)
    What exactly is your issue with 'this'

    (Original post by Zenarthra)
    I dont have an issue but is it correct?
    Why have some members told me to use a PI of axsin(3x)+bx(sin3x)?
    when f(x)=4sin(3x) is contained in the cf?

    (Original post by FeelsToWaltz)
    Yeah this is confusing me too as I've never seen a question on it. Judging by the textbook, if the Complementary Function is acosx + bsinx and the right hand side is for example 2cosx then the PI is axsinx, and if the right hand side is 2sinx then the PI is axcosx.

    Really hoping they don't give us a question on this tomorrow as I have never used it before!
    Well there you go, I'd never noticed that - it's entirely true, if you try differentiating and substituting. I've spent so much time making sure that my students use the cos term and the sin term when the x is not needed, that I'd never noticed that you can simplify it in the case with the x in. However, if you use both terms, it will all turn out OK.
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    (Original post by davros)
    I'd go with that PI myself.

    (This is really something you should be able to test out for yourself! Make up a DE with the type of f(x) where you have an issue; try the different possible PIs that you are debating, and see what comes out. You should be able to see from the end result which type of PI is going to work and which one is going to give you problems because you either end up with 0 = f(x) or something like acos(mx) + bxsin(mx) = Asin(mx) which can't be solved.)
    We havent had a question of this type.
    DO you have any links that have these types of questions so i can test it out?
    Would prefer to do a set question rather than making one up for myself.
    Unless you want to for us.
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    (Original post by tiny hobbit)
    Well there you go, I'd never noticed that - it's entirely true, if you try differentiating and substituting. I've spent so much time making sure that my students use the cos term and the sin term when the x is not needed, that I'd never noticed that you can simplify it in the case with the x in. However, if you use both terms, it will all turn out OK.
    Did you try this for the example given?
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    (Original post by Zenarthra)
    We havent had a question of this type.
    DO you have any links that have these types of questions so i can test it out?
    Would prefer to do a set question rather than making one up for myself.
    Unless you want to for us.
    Just make one up - it's a useful exercise!

    You know what you have to do:

    if you want a CF with something like sin(3x) in it then you need an AE that looks like m^2 + 9 = 0, Now choose an f(x) of a similar form e.g. 2sin(3x) or 5cos(3x) or 4sin(3x) + 3cos(3x). Does a PI with just an xsin(3x) work? Does a PI with just an xcos(3x) work? What problems do you run into? Can you make a PI work with Axsin(3x) + Bxcos(3x)?
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    (Original post by Phichi)
    Did you try this for the example given?
    Yes. See Davros' posting above.
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    (Original post by tiny hobbit)
    Yes. See Davros' posting above.
    I'm not looking for help, I was looking to see whether it worked for you, I'm lazy. Far into uni now!
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    (Original post by Phichi)
    I'm not looking for help, I was looking to see whether it worked for you, I'm lazy. Far into uni now!
    I'm in a minority on this forum. I only state things if I'm sure!

    If I'm not sure, I'll make that quite clear.
 
 
 
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