Using Galerkin's method, solve the following differential equation with boundary values,

doing what is shown below.

$h^e{h}^e=1y(1)y^{\text{'}}=d\frac{y}{d}xx=0x=2(0,2)y=\frac{x_{x-2}}{2}{N}_1=\frac{1}{2}(1+)$

$N_2=\frac{1}{2}(1-)$

$dx=\frac{h_e}{2}d\frac{d^{2}y}{d{x}^2}=1\mathrm{.}0$;$0$

Express the weak form $of$ the problem.

Using isoparametric linear interpolation, find the matrix $of$ "stiffness" for a particular

element $of$ length $h^e.$

Assemble the global stiffness matrix assuming two linear elements $of$ two nodes

long $h^e=1.$

Solve the resulting system $of$ equations and find the discrete values $iny(1).$

Find the discrete value $of$ the derivatives $y^{\text{'}}=d\frac{y}{d}xinx=0$ and $x=2.$

Graph the solution obtained $in$ the interval $of(0,2)$ assuming the variation assumed

linear and compare $it$ with the exact solution which $isy=\frac{x_{x-2}}{2.}$

Tips: The shape functions $to$ use are:

$N_1=\frac{1}{2}(1+)$

$N_2=\frac{1}{2}(1-)$

$dx=\frac{h_e}{2}d$