# Question:Piecewise Definition of a Green's Function for a Fourth-Order Differential Equation

## Question:Piecewise Definition of a Green's Function for a Fourth-Order Differential Equation

Maple

I have just begun to study Green's functions and made some initial progress on a problem, but now need Maple to make further progress.  Apologies, I have written up the equations in LaTeX form rather than Maple, as my Maple has gotten very rusty.

$\frac{d^4y}{dx^4}=f(x) y(0)=y'(0)=0, y(1)=y'(1)=0$

I showed that the Green function $G(x,u)$ for this equation satisfies a condition

$\lim_{\epsilon\to0}\bigg[\frac{\partial^3G}{\partial x^3}\bigg]_{x=u-\epsilon}^{u+\epsilon}=1$

and showed that there is continuity of the Green's function and its first and second partial derivatives with respect to $x$ at $x=u$.  The next step is to show that this function has a piecewise definition such that

$G=\frac{1}{6}x^2(1-u)^2(3u - 2ux -x$ for x between the range 0 and u and such that

\[G(x,u)=\frac{1}{6}u^2(1-x^2)(3x - 2xu - u) for x between u and 1

I am not entirely sure how to do this with pen and paper, so I have reason to believe that it could be a done a lot more easily with Maple, if someone could give some pointers that would be much appreciated.

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