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Math Help - find constants D and E when given the derivative

  1. #1
    Junior Member
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    Oct 2008
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    34

    find constants D and E when given the derivative

    let g(x) = e^(-4x) * (Dsin3x + Ecos3x)
    where D and E are constants. Find values D and E if
    g'(x) = e^(-4x) * sin3x

    Here's my work:

    *must find the derivative and then equate to e^(-4x) * sin3x
    g'(x) = [e^(-4x) * -4* (D*cos3x + E*sin3x)] + [e^(-4x) * (3D*cos3x - 3E*sin3x)]

    simplified, and factored
    g'(x) = (e^(-4x)) * [(-4D-3E)(sin3x) + (3d-4E)(cos3x)]

    so now we equate to the derivative from the equation
    (e^(-4x)) * [(-4D-3E)(sin3x) + (3D-4E)(cos3x)] = e^(-4x) * sin3x

    we can assume that
    (3D-4E) = 0 and (-4D-3E) = 1


    THIS is where i'm stuck. I know i need to find the value of one in order to find the value of the other constant value, but HOW?!
    in previous problems we would have a third constant that normally worked out to be set as something, but in this we dont.
    Either i messed up big time in my work or I'm missing something quite obvious.

    Either way please please help
    Thanks,
    Brittany
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  2. #2
    Senior Member
    Joined
    Dec 2008
    Posts
    319
    You are correct that

    \begin{aligned}<br />
3D-4E &= 0\\<br />
-4D-3E &= 1.<br />
\end{aligned}

    To solve for D and E, try substitution:

    \begin{aligned}<br />
3D-4E&=0\\<br />
3D&=4E\\<br />
D&=\frac{4}{3}E.<br />
\end{aligned}
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