The variable $u(x),$ e$.$g$.,$ temperature, through a $1-$D region is governed by:

$a=b=1,L=0\mathrm{.}5m{u}^{**}(x)=cx(L^2-x^2)u(\frac{L}{2})\frac{d^{2}u}{d{x}^2}+b{x}^2+a=0,0$ Data: $a=b=1,L=0\mathrm{.}5m$

Assume the trial approximation function $u^{**}(x)=cx(L^2-x^2).$

$(a)-$Determine the unknown parameter $c$ via the Galerkin $WRM.$

$(b)-$ Predict $uat$ the mid$-$span, $u(\frac{L}{2}).$

Solution $-$ show work!