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Math Help - Help with nonhomogeneous second order Diff. Eq.

  1. #1
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    Help with nonhomogeneous second order Diff. Eq.

    Hello,

    I am currently working on solving nonhomogeneous differential equations using the method of undetermined coefficients. Im doing well solving problems, but I have come across one that has puzzled me, possibly due to its simplicity.

    The initial value problem is:
     y'' + y' - 2y = 2t, ~ ~ y(0) = 0, ~ ~ y'(0) = 1

    I have found the complementary general solution,
     y(t) = c_1e^t + c_2e^{-2t}
    but before I can solve for the constants, I must account for the nonhomogeneous term.

    I make my first assumption that the term is of the form:
     Y(t) = At
    therefore
     Y'(t) = A,
     Y''(t) = 0
    <br />
A - 2At = 2t
    Obviously this is not a solution so I try the only other thing I know to try.
     Y(t) = At^2
     \therefore  Y'(t) = 2At,
     \ Y''(t) = 2A
    so
     2A + 2At - 2At^2 = 2t
    This isnt going to work either and now Im out of ideas. Any help is greatly appreciated.
    Last edited by sixstringartist; October 23rd 2007 at 03:59 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by sixstringartist View Post
    Hello,

    I am currently working on solving nonhomogeneous differential equations using the method of undetermined coefficients. Im doing well solving problems, but I have come across one that has puzzled me, possibly due to its simplicity.

    The initial value problem is:
     y'' + y' - 2y = 2t, ~ ~ y(0) = 0, ~ ~ y'(0) = 1

    I have found the complementary general solution,
     y(t) = c_1e^t + c_2e^{-2t}
    but before I can solve for the constants, I must account for the nonhomogeneous term.

    I make my first assumption that the term is of the form:
     Y(t) = At
    therefore
     Y'(t) = A,
     Y''(t) = 0
    <br />
A - 2At = 2t
    Obviously this is not a solution so I try the only other thing I know to try.
     Y(t) = At^2
     \therefore Y'(t) = 2At,
     \ Y''(t) = 2A
    so
     2A + 2At - 2At^2 = 2t
    This isnt going to work either and now Im out of ideas. Any help is greatly appreciated.
    Try: Y(t)=At+B

    RonL
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  3. #3
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     y'' + y' -2y = 2t, ~~ y(0) = 0, ~~ y'(0) = 1
     Y(t) = At + B
     Y'(t) = A
     Y''(t) = 0
     A - 2At - 2B = 2t
    -2A = 2, ~~ A-2B = 0
    A = -1, ~~ B = \frac {-1}{2}

    And that is the correct coefficients. Thank you very much.
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