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Math Help - Variation of Parameters

  1. #1
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    I have a test coming up involving variation of parameters and I am practicing. I got stuck on this one, I know it should just be algebra but now I'm lost.

    Using the system of equations:

    U1'(y1)+U2'(y2) = 0
    U1'(y1')+U2'(y2') = f(x)

    I need to solve for U1' and U2'.

    I started with y" + 16y = 4Tan(4t)

    I found that r = 0 (+-) 4i, and y(c) = Cos(4t) + Sin(4t)

    using that fact I did what was necessary to enter that into the system of equations and now I can't figure out what I need to do in order to remove U1' and U2' from the equation separately in order to figure out what U2' and U1' are.

    Any suggestions of what I should do next? What to multiply/divide/subtract/add/etc.


    Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?

    -WWB
    Last edited by mr fantastic; April 18th 2009 at 04:10 PM. Reason: Merged posts
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  2. #2
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    Quote Originally Posted by Whitewolfblue View Post
    I have a test coming up involving variation of parameters and I am practicing. I got stuck on this one, I know it should just be algebra but now I'm lost.

    Using the system of equations:

    U1'(y1)+U2'(y2) = 0
    U1'(y1')+U2'(y2') = f(x)

    I need to solve for U1' and U2'.

    I started with y" + 16y = 4Tan(4t)

    I found that r = 0 (+-) 4i, and y(c) = Cos(4t) + Sin(4t)

    using that fact I did what was necessary to enter that into the system of equations and now I can't figure out what I need to do in order to remove U1' and U2' from the equation separately in order to figure out what U2' and U1' are.

    Any suggestions of what I should do next? What to multiply/divide/subtract/add/etc.


    Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?

    -WWB

    you should be removing the u primes from the equation and setting them equal to zero, and taking the derivative of whatever is left.

    Afterwards you use the u primes as your system of equations, a bunch of stuff should cancel, if not you did something wrong.

    you should be able to solve for all the primes individually as a system of equations or using the wronskian, although depending the wronskian can be much simpler.
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  3. #3
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    What about this question as well? I'd rather not use wrongskian. Our instructor specifically requested we learn how to use VOP fully.

    "Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?"

    I think what I did was put the Y(c) into both y1 and y2.

    -WWB
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  4. #4
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    Quote Originally Posted by Whitewolfblue View Post
    What about this question as well? I'd rather not use wrongskian. Our instructor specifically requested we learn how to use VOP fully.

    "Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?"

    I think what I did was put the Y(c) into both y1 and y2.

    -WWB
    no you leave the homogenous solution as a whole. You just differentiate the whole thing set all your u primes to zero and solve. Whatever is left is part of the particular.

    Theres nothing really fully about it, and using the wrongskian is merely a place holder which saves time instead of solving it as a system of equations which ends up being the same thing because its a matrix.

    you're y(c) is your homogenous, just differentiate and solve that using your system.

    unless im misunderstanding your original question...
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  5. #5
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    Lightbulb Got it

    Quote Originally Posted by p00ndawg View Post
    no you leave the homogenous solution as a whole. You just differentiate the whole thing set all your u primes to zero and solve. Whatever is left is part of the particular.

    Theres nothing really fully about it, and using the wrongskian is merely a place holder which saves time instead of solving it as a system of equations which ends up being the same thing because its a matrix.

    you're y(c) is your homogenous, just differentiate and solve that using your system.

    unless im misunderstanding your original question...
    I see what your talking about now.

    Thanks,

    -WWB
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