Solve the following ODE, which describes the steady state temperature distribution of a

rod, with the Finite Difference Method.

$\frac{d^{2}T}{d{x}^2}+0\mathrm{.}5(10-T)=0$;$,T(0)=200,T(10)=100$

Use the $2^{nd}$ order accurate center difference scheme with a step size $x=0\mathrm{.}01$ to

discretize the derivative term.

$($a$)$ Provide the FDE at the $($i$)$ left$-$most, $($ii$)$ right$-$most, and $($iii$)$ an interior point.

$($b$)$ Generate a MATLAB script that creates the matrix resulting from the linear system

of algebraic equations. Use your existing LU decomposition $O(N^3),$ AND, incorporate

the provided Thomas Algorithm $O(N)$ function to solve the system of equations.

Compare the solve times and provide both the code and the output $($plotting $T$ vs$.x).$