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Math Help - Two Webwork Questions (Second order)

  1. #1
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    Two Webwork Questions (Second order)

    I've been attempting these two questions a few times and I countinue to get them wrong.

    1) Find y as a function of x if
     x^2(y'') -15xy'  + 64y = x^3 y(1)=6, y'(1)=-5
    So I solved the Homogenous equation as a Euler equation. My homogeneous solution is yh= C1x^(8)+C2ln(x)x^(8)
    then i proceeded with undetermined coefficients and ended up with 151/25x^8-1336/25ln((x))x^8+x^(3)/25
    Whats wrong with the method?

    2) Use the method of undetermined coefficients or the method of differential operators to find one solution of
    y'' -8y' + 43y = 32e^4t(cos(5t))+64e^4t(sin(5t))+ 7e^0t
    (It doesn't matter which specific solution you find for this problem.)
    My guess for yp was Ae^4t cos(5t)+Be^ 4t sin(5t)
    And I used the undetermined coefficent method to find A=16 and B=32.

    Thanks.
    Last edited by electricalphysics; October 25th 2009 at 03:47 PM.
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  2. #2
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    As to (1), what did you get as your specific solution to the non-homogeneous equation (to which you added the general homogeneous solution)?

    --Kevin C.
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  3. #3
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    for (1) my specific solution was y=x^(3)/25
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  4. #4
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    anyone have any idea whats wrong ?
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  5. #5
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    Your specific solution seems to be correct, as does your general homgeneous solution. Thus, you have general solution y(x)=(C_1+C_2\ln{x})x^8+\frac{x^3}{25}.
    Setting y(1)=6, we get
    6=(C_1+C_2\ln(1))1^8+\frac{1^3}{25}
    6=(C_1+C_2\cdot0)+\frac{1}{25}
    6=C_1+\frac{1}{25}
    C_1=6-\frac{1}{25}=\frac{149}{25}
    you have C_1=\frac{151}{25}=6+\frac{1}{25}, which by the above, you can see is incorrect.
    With C_1=\frac{149}{25}, set y'(1)=-5, and solve for the correct C_2.

    --Kevin C.
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  6. #6
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    As to (2),note that 7e^{0t}=7e^{0}=7, so that your differential equation is
    y''-8y'+43y=32e^{4t}\cos{5t}+64e^{4t}\sin{5t}+7e^{0t}
    which is
    y''-8y'+43y=32e^{4t}\cos{5t}+64e^{4t}\sin{5t}+7
    what you gave as a solution is in fact a solution to
    y''-8y'+43y=32e^{4t}\cos{5t}+64e^{4t}\sin{5t},
    not the above equation.
    [Hint: consider what happens when you add a constant to the solution you have].

    --Kevin C.
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