Uniform beams under a compressive load $P$ and distributed transverse load $w(x)$ satisfy the fourth$-$order ODE

$\frac{d^{4}y}{d{x}^4}+\frac{P}{EI}\frac{d^{2}y}{d{x}^2}=\frac{1}{EI}w(x)$

where $y(x)$ is the deflection of the beam as a function of the position $x$ along the length of the beam. Consider such a beam with ends at $x=0$ and $x=L,$ both of which are hinged. In this problem you will find the Green function for the beam, and use it to find the shape of the beam under a uniformly$-$distributed transverse load.

$($a$)$ Find the solution family for the homogeneous equation

$\frac{d^{4}g}{d{x}^4}+{}^2\frac{d^{2}g}{d{x}^2}=0$

where ${}^2=\frac{P}{E}$I for convenience. Write the solution family in terms of real functions, and make sure that you have four arbitrary constants present in the solution.

$($b$)$ Apply the hinged constraints $g(0)=0$ and $g^{\text{'}\text{'}}(0)=0$ to your solution family from part $($a$).$ You should be left with only two arbitrary constants present in the solution. $($Rename these $C_1$ and $C_2)$

$($c$)$ Define your Green function to be

$G(x$;$y)=\{\begin{array}{ccc}{G}_L(x)& if& xy\end{array}$

where $G_L(x)=g(x)$ from part $($b$)$ with the constants renamed to $C_{L1}$ and $C_{L2},$ and $G_R(x)=g(L-x)$ from part $($b$)$ with the constants renamed to $C_{R1}$ and $C_{R2}.$

$($d$)$ Apply the continuity$/$discontinuity conditions

$G_L(y)=G_R(y),G_L^{\text{'}}(y)=G_R^{\text{'}}(y),G_L^{\text{'}\text{'}}(y)=G_R^{\text{'}\text{'}}(y),G_L^{\text{'}\text{'}\text{'}}(y)+1=G_R^{\text{'}\text{'}\text{'}}(y)$

and solve for the values of $C_{L1},C_{L2},C_{R1},$ and $C_{R2}.$

$($e$)$ Use the integral

$y(x)=\frac{1}{EI}{\displaystyle \underset{0}{\overset{L}{}}}G(x$;$y)w(y)dy$

$=\frac{1}{EI}{\displaystyle \underset{0}{\overset{x}{}}}{G}_R(x$;$y)w(y)dy+\frac{1}{EI}{\displaystyle \underset{x}{\overset{L}{}}}{G}_L(x$;$y)w(y)dy$

to solve for the deflection of the beam when $w(x)=g,$ where $$ and $g$ are constants.