$1\mathrm{.}6-1\mathrm{.}9$ Obtain an approximate displacement equation for the simply supported beam shown in Figure Pl$.6$ using the trial solution $y(x)=$Asin$\frac{x}{H}.$ Compare the deflection at the center with the theoretical value $y=-0\mathrm{.}06415M_0\frac{H^2}{E}$I. The governing differential equation is

$EI\frac{d^{2}y}{d{x}^2}-\frac{M_{0}x}{H}=0$

Figuro P$1.6$

$1\mathrm{.}6$ Evaluate $A$ by minimizing the integral

$={\displaystyle \underset{0}{\overset{H}{}}}[\frac{EI}{2}(\frac{dy}{dx}{)}^2+\frac{M_{0}x}{H}y]dx$

$1\mathrm{.}7$ Evaluate $A$ by requiring the residual to vanish at $($a$)x=\frac{H}{2},$ and $($b$)x=0\mathrm{.}577H.$