For a thick pressure vessel of inner radius, a and outer radius $b,$ the differential equation for the radial displacement $u$ of a point along the thickness is given by:

$($I$)$

$\frac{d^{2}u}{d{r}^2}+\frac{1}{r}\frac{du}{dr}-\frac{u}{r^2}=0$

Using finite difference approximations

$\frac{d^{2}u}{d{r}^2}$~~$\frac{u_{i+1}-2u_i+u_{i-1}}{(r{)}^2}$

$\frac{du}{dr}$~~$\frac{u_{i+1}-u_i}{r}$

Develop the discretized equations for displacement of equation I

Using $a=5^{\text{'}\text{'}}$ and $b=8^{\text{'}\text{'}}$ using $r={0\mathrm{.}6}^{\text{'}\text{'}}$ develop the system of equations for estimating displacements $($

Strains$)$ at each node. $($number of nodes $n=\frac{b-a}{}r=\frac{8-5}{0\mathrm{.}6}=5$

Solve the system of equations using Gauss Elimination methods to get the stress at each node. $($ show your work in detail$)$