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22 March, 09:27

Which of the following functions are solutions of the differential equation y+4y+4y=0? A. y (x) = e^-2x B. y (x) = e^+22 C. y (x) = xe^-2x D. y (x) = - 2x E. y (x) = 0 F. g (x) = x^2e^-2x

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  1. 22 March, 09:46
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    Answer: E. y (x) = 0

    Step-by-step explanation:

    y (x) = 0 is the only answer from the options that satisfies the differential equal y" - 4y' + 4y = 0

    See:

    Suppose y = e^ (-2x)

    Differentiate y once to have

    y' = - 2e^ (-2x)

    Differentiate the 2nd time to have

    y" = 4e^ (-2x)

    Now substitute the values of y, y', and y" into the give differential equation, we have

    4e^ (-2x) - 4[-2e^ (-2x) ] + 4e^ (-2x)

    = 4e^ (-2x) + 8e^ (-2x) + 4e^ (-2x)

    = 16e^ (-2x)

    ≠ 0

    Whereas we need a solution that makes the differential equation to be equal to 0.

    If you test for the remaining results, the only one that gives 0 is 0 itself, and that makes it the only possible solution from the options.

    It is worth mentioning that apart from the trivial solution, 0, there is a nontrivial solution, but isn't required here.
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