Dynamics of a beam

**pdnhan** · May 16th 2011, 11:55 PM

Relevant to the dynamics of a beam, consider the fourth order linear differential operator.
$L=\frac{d^4}{dx^4}$
a. Show that $uLv - vLu$ is an exact differential. It is a differential of an expression with four terms, each term a product of $u$ and $v$ or their derivatives.
b. Evaluate $\int \begin{array}{cc}1\\0\end{array} (uLv - vLu) dx$ in terms of boundary data for any functions $u$ and $v$ .
Please help

**Danny** · May 17th 2011, 06:34 AM

This should get you started

$u \frac{d^4v}{dx^4} - v \frac{d^4u}{dx^4} = \dfrac{d}{dx} \left(u \frac{d^3v}{dx^3} - v \frac{d^3u}{dx^3} - \frac{du}{dx} \frac{d^2v}{dx^2}+ \frac{dv}{dx} \frac{d^2u}{dx^2}\right)$

**pdnhan** · May 18th 2011, 06:09 AM

thank you

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