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20 September, 21:10

A. By inspection, find a particular solution of y'' + 2y = 6. yp (x) =

(b) By inspection, find a particular solution of y'' + 2y = - 12x. yp (x) =

(c) Find a particular solution of y'' + 2y = - 12x + 6. yp (x) =

(d) Find a particular solution of y'' + 2y = 24x + 3. yp (x) =

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  1. 20 September, 22:00
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    (a) The particular solution, y_p is 3

    (b) y_p is - 6x

    (c) y_p is - 6x + 3

    (d) y_p is 12x + 3

    Step-by-step explanation:

    To find a particular solution to a differential equation by inspection - is to assume a trial function that looks like the nonhomogeneous part of the differential equation.

    (a) Given y'' + 2y = 6.

    Because the nonhomogeneus part of the differential equation, 6 is a constant, our trial function will be a constant too.

    Let A be our trial function:

    We need our trial differential equation y''_p + 2y_p = 6

    Now, we differentiate y_p = A twice, to obtain y'_p and y''_p that will be substituted into the differential equation.

    y'_p = 0

    y''_p = 0

    Substitution into the trial differential equation, we have.

    0 + 2A = 6

    A = 6/2 = 3

    Therefore, the particular solution, y_p = A is 3

    (b) y'' + 2y = - 12x

    Let y_p = Ax + B

    y'_p = A

    y''_p = 0

    0 + 2 (Ax + B) = - 12x

    2Ax + 2B = - 12x

    By inspection,

    2B = 0 = > B = 0

    2A = - 12 = > A = - 12/2 = - 6

    The particular solution y_p = Ax + B

    is - 6x

    (c) y'' + 2y = - 12x + 6

    Let y_p = Ax + B

    y'_p = A

    y''_p = 0

    0 + 2 (Ax + B) = - 12x + 6

    2Ax + 2B = - 12x + 6

    By inspection,

    2B = 6 = > B = 6/2 = 3

    2A = - 12 = > A = - 12/2 = - 6

    The particular solution y_p = Ax + B

    is - 6x + 3

    (d) Find a particular solution of y'' + 2y = 24x + 3

    Let y_p = Ax + B

    y'_p = A

    y''_p = 0

    0 + 2 (Ax + B) = 24x + 6

    2Ax + 2B = 24x + 6

    By inspection,

    2B = 6 = > B = 6/2 = 3

    2A = 24 = > A = 24/2 = 12

    The particular solution y_p = Ax + B

    is 12x + 3
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