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Math Help - Undetermined Coefficients

  1. #1
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    Undetermined Coefficients

    Solve the initial value problem using undetermined coefficients

    2y" + 3y' - 2y = 14x^2 - 4x - 11 ; y(0)=0 ,  y'(0)=(0)
    Use auxiliary to solve for Yc = c1e^(^1^/^2^)^x + c2e^-^2^x

    Assume Yp = Ax^2 + Bx + C
    Yp' = 2Ax + B
    Yp" = 2A
    Plug into original equation:
    4A + 6Ax + 3B - 2Ax^2 - 2Bx - 2C = 14x^2 - 4x - 11
    -2Ax^2 = 14x^2
    A = -7

    6Ax - 2Bx = -4x
    B = -19

    4A + 3B - 2C = -11
    C=-37

    So Yp = -7x^2 - 19x - 37

    Y = c1e^(^1^/^2^)^x + c2e^-^2^x - 7x^2 - 19x - 37
    Y' = (1/2)c1e^(^1^/^2^)^x - 2c2e^-^2^x - 7x - 19
    Y(0) = c1 + c2 - 37 = 0
    c1 = 37 - c2
    Y'(0) = (1/2)c1 - 2c2 - 19 = 0

    (1/2)(37 - c2) - 2c2 = 19
    c2 = -(1/5)
    c1 = 186/5

    Y = (186/5)e^(^1^/^2^)^x - (1/5)e^-^2^x - 7x^2 - 19x - 37

    I'd appreciate it if someone could check through my work and make sure I didn't make an error somewhere, thanks in advance.
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  2. #2
    Master Of Puppets
    pickslides's Avatar
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    You can check it yourself by taking the derivative twice and substituting these results back into the original equation.

    Is the original equation satisfied?
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