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Math Help - Finding general solution

  1. #1
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    Finding general solution

    Given that three solutions to the inhomongenous DE equation y''+p(t)y'+q(t)y=g(t)

    are y1=1+e^(t^2), y2= 1+te^(t^2), y3= 1+(t+1)e^(t^2),

    find the general solution
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  2. #2
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    Since y_1, y_2 and y_3 all satisfy the ODE then

    u_1 = y_3 - y_2 = e^{t^2} and u_2 = y_3 - y_2 = t e^{t^2}

    satisfy

    u'' + pu' + q u = 0

    Thus, the general solution is

    y = c_1 e^{t^2} + c_2 t e^{t^2} + 1.

    Note that we could have used either y_1, y_2\; \text{or}\; y_3 for y_p but can adjust the constants such that all we need is y_p =1.
    Last edited by Jester; May 21st 2011 at 01:18 PM.
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  3. #3
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    Why did you subtract the solutions? Thanks
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  4. #4
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    It was to get to the homogeneous ODE

    If y_1 and y_2 satisfy your ODE, i.e.

    y_1'' + py_1' + q y_1 = g

    and

    y_2'' + py_2' + q y_2 = g

    then substracting gives

    (y_2''-y_1'') + p(y_2'-y_1') + q(y_2-y_1) = 0.
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