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Math Help - differential equation

  1. #1
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    differential equation

    A nonhomogeneous differential equation, a complementary solution
    yc and particular solution yp are given. Find the solution satisfying the given initial conditions.
    y′′ 4y = 12; y(0) = 0, y(0) = 10; yc = c1e^2x + c2e^(2x), yp = 3
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  2. #2
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    Hello, Harry!

    A nonhomogeneous differential equation.
    A complementary solution y
    c and particular solution yp are given.

    Find the solution satisfying the given initial conditions.

    . . y′′ − 4y .= .12, . y
    c .= .C1e^{2x} + C2e^{-2x}, .yp = -3

    . . y(0) = 10, .y'(0) = 10
    We have: . y .= .C1e^{2x} + C2e^{-2x} - 3


    We are told: .y(0) = 0
    . . Hence: we have: .C
    1e^0 + C2e^0 - 3 .= .0 . . C1 + C2 .= .3 .[1]

    We are told: .y'(0) = 10
    . . y' .= .2C
    1e^{2x} - 2C2e^{-2x}
    . . Hence: .2C
    1e^0 - 2C2e^0 .= .10 . . C1 - C2 .= .5 .[2]

    Add [1] and [2]: .2C
    1 = 8 . . C1 = 4

    Substitute into [1]: .C
    2 = -1


    Therefore: . y .= .4e^
    {2x} - e^{-2x} - 3

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