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Thread: Undetermined Coefficients

  1. #1
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    Undetermined Coefficients

    Solve the initial value problem using undetermined coefficients

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

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

    $\displaystyle 6Ax - 2Bx = -4x$
    $\displaystyle B = -19$

    $\displaystyle 4A + 3B - 2C = -11$
    $\displaystyle C=-37$

    So $\displaystyle Yp = -7x^2 - 19x - 37$

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

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

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

    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
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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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