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Math Help - [SOLVED] Can't find particular solution of nonhomogenous linear ode

  1. #1
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    [SOLVED] Can't find particular solution of nonhomogenous linear ode

    My problem:

    y'' + 4y' + 5y = 5t + e^{-t}

    I found the general solution of the homogenous equation

    r^2 + 4r + 5 = 0

    to be

    Y_c = C_1e^{-2t}cos(t) + C_2e^{-2t}sin(t)

    since

    r = -2 + i, -2 - i (complex)


    Now, I have tried all of the following as particular solutions and all have come up 'no solution'

    Be^{-t}
    Bte^{-t}
    A + Be^{-t}
    A + Bte^{-t}
    At + Be^{-t}
    At + Bte^{-t}
    At^2 + Be^{-t}
    At^2 + Bte^{-t}
    At^2 + Bt^2e^{-t}
    At^3 + Be^{-t}
    At^3 + Bte^{-t}
    At^3 + Bte^{-t}
    At^3 + Ct^2 + Dt + Be^{-t}

    All of them come up with conflicting values (A will equal 0 in one spot and need to be a value in another).

    Can anyone tell what I am doing wrong?

    thanks
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  2. #2
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    Your complementary solution is fine.

    Now for the particular solution:

    y_p = A + Bt + De^{-t}

    y_p' = B - De^{-t}

    y_p'' = De^{-t}.


    Therefore

    y'' + 4y' + 5y = De^{-t} + 4(B - De^{-t}) + 5(A + Bt + De^{-t})

     = De^{-t} + 4B - 4De^{-t} + 5A + 5Bt + 5De^{-t}

     = 5A + 4B + 5Bt + 2De^{-t}.


    Since this is equal to 5t + e^{-t} that means

    5A + 4B = 0, 5B = 5, 2D = 1

    Solving simultaneously gives

    A = -\frac{4}{5}, B = 1, D = \frac{1}{2}.

    So y_p = -\frac{4}{5} + t + \frac{1}{2}e^{-t}.



    So finally, the general solution is

    y_g = y_c + y_p

     = C_1e^{-2t}\cos{t} + C_2e^{-2t}\sin{t} - \frac{4}{5} + t + \frac{1}{2}e^{-t}.
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  3. #3
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    Quote Originally Posted by Prove It View Post
    Your complementary solution is fine.

    Now for the particular solution:

    y_p = A + Bt + De^{-t}

    y_p' = B - De^{-t}

    y_p'' = De^{-t}.

    I was so close! I need to be a little more careful with my guesses and more methodical in my choices.

    Using the method of undetermined coefficients was presented to me in a guess & check manner. Is there a better way? This is too tedious and the longer it takes to find a solution, the more my confidence in my work subsides.

    thank you for your help!
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  4. #4
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    The undetermined coefficients methods is exactly that - guess and check.

    If you have a polynomial, you would expect your particular solution to have a polynomial of the same degree. Of course, that means you need coefficients for each power and a constant.

    If you have a sinusoidal function, you would expect your particular solution to be a sum of a sine and cosine function (with the same multiple angle).

    If you have an exponential function, you would expect your particular solution to have an exponential function of the same power.

    If you have any sum of the above types of functions, your particular solution will have a sum of these types of functions.

    If you have a function that already appears in your complementary solution, you will need to make a guess of that function multiplied by the independent variable.


    I suggest you have a read of http://www.mathhelpforum.com/math-he...-tutorial.html
    as it explains these in more detail, and also gives other methods of solving second order DEs (Annihilator Method and Variation of Parameters).
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