= Clamped-beam Green function
{title2=$G(x,\xi)=x_<^2(1-x_>)^2[3x_>-(1+2x_>)x_<]/6$}
For a unit-length <elastic beam> with unit bending stiffness, the <Green function> of $D_x^4$ with $G=G_x=0$ at both endpoints is the displayed kernel. It is piecewise cubic and continuous through its second <derivative>, while $G_{xxx}(\xi+,\xi)-G_{xxx}(\xi-,\xi)=1$. A constructive form is $G=ax^2+bx^3+(x-\xi)_+^3/6$, with $a=\xi(1-\xi)^2/2$ and $b=-(1-\xi)^2(1+2\xi)/6$. The right endpoint conditions determine these coefficients. Thus $y(x)=\int_0^1G(x,\xi)f(\xi)d\xi$ solves the clamped static beam problem.
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