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2 February, 17:45

Find a particular solution to the differential equation using the Method of Undetermined Coefficients.

9y'' + 5y' - y = 14

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  1. 2 February, 20:22
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    y = c1e^[ (-5 + √61 (/18] + c2e^[ (-5 - √61) / 18] - 14

    Step-by-step explanation:

    To solve the differential equation

    9y'' + 5y' - y = 14 (equation 1)

    is to obtain the general equation y = y_c + y_p.

    To do that, we need the complimentary function, y_c, and the particular integral, y_p.

    To obtain the particular integral, using the method of undetermined coefficients, we need a trial function, y_p, such that

    9y_p'' + 5y_p' - y_p = 14 (equation 2)

    This works by guessing. When a guess gives a trivial y_p = 0, we then make another guess.

    14 is a constant, we guess a constant trial function.

    Let y_p = A

    y_p' = 0

    y_p'' = 0

    Substitute these values into equation 2

    9 (0) + 5 (0) - (0*x + B) = 14

    - B = 14 or B = - 14

    y_p = Ax + B = - 14 (equation 3)

    To obtain the compliment function, we need to solve the homogeneous part of equation 1.

    The homogeneous part is

    9y'' + 5y' - y = 0

    The auxiliary equation is

    9m² + 5m - 1 = 0

    Solving the quadratic equation, using the quadratic formula

    x = [-b ± √ (b² - 4ac) ]/2a

    With a = 9, b = 5, and c = 0

    x = [-5 ± √ (5² - 4*9 * (-1)) ]/2 (9)

    = [-5 ± √ (25 + 36) ]/18

    x1 = (-5 + √61) / 18

    x2 = (-5 - √61) / 18

    y_c = c1e^ (x1) + c2e^ (x2)

    y_c = c1e^[ (-5 + √61 (/18] + c2e^[ (-5 - √61) / 18]

    The general solution

    y = c1e^[ (-5 + √61 (/18] + c2e^[ (-5 - √61) / 18] - 14
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