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Thread: 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 $\displaystyle y_1$, $\displaystyle y_2$ and $\displaystyle y_3$ all satisfy the ODE then

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

    satisfy

    $\displaystyle u'' + pu' + q u = 0$

    Thus, the general solution is

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

    Note that we could have used either $\displaystyle y_1, y_2\; \text{or}\; y_3$ for $\displaystyle y_p$ but can adjust the constants such that all we need is $\displaystyle y_p =1$.
    Last edited by Jester; May 21st 2011 at 12:18 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 $\displaystyle y_1$ and $\displaystyle y_2$ satisfy your ODE, i.e.

    $\displaystyle y_1'' + py_1' + q y_1 = g$

    and

    $\displaystyle y_2'' + py_2' + q y_2 = g$

    then substracting gives

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