Q$2$: ODE eigenvalue problem

An axially loaded wood column $($simply supported on both ends$)$ has the following characteristics:

$-\backslash ($ E$=10\backslash$times $10^\{9\}[\backslash$mathrm$\{$~Pa$\}]\backslash )$

$-\backslash ($ I$=1\mathrm{.}25\backslash$times $10^\{-5\}\backslash$left$[$m$^\{4\}\backslash$right$]\backslash )$

$-\backslash ($ L$=3[$m$]\backslash )$

$\backslash [$

P$=\backslash$pi$^\{2\}\backslash$frac$\{$E I$\}\{$L$^\{2\}\},$

$\backslash ]$

where $\backslash ($ P $\backslash )$ is the analytical solution for the critical buckling load.

Reference Equations $27\mathrm{.}17,27\mathrm{.}18,27\mathrm{.}20,$ and Examples $27\mathrm{.}7$ and $27\mathrm{.}8$ of the textbook for this question.

$1.(1$ point$)$ Calculate the analytical buckling load of the first mode $(\backslash (\backslash$mathrm$\{$n$\}=1\backslash ))$ using Equation $1.$

$2.(2$ points$)$ By coding, use Power Method in Matlab or Python which takes two inputs: the matrix and the number of iterations.

The code should return the smallest eigenvalue. Submit your code.

$($Hint $01$: You can use the same code as Q$1,$ either the AI$-$generated one or your own one.$)$

$($Hint $02$: You can use built$-$in functions in Matlab or Python to find the matrix inverse.$)$

$3.(3$ points$)$ Using finite differences $[$see Equation $27\mathrm{.}18$ of the textbook$],$ set up the coefficient matrix that results from using $5$ nodes $(2$ boundary nodes and $3$ interior nodes$),$ evenly distributed along the column. By calling your Power Method function, compute the buckling load after $1,2,3,4,$ and $5$ iterations. Submit your code, the tabulated results of the numerically$-$approximated buckling load vs$.$ Power Method iterations.

$4.(4$ points$)$ Increase BOTH the level of discretization $($i$.$e$.,$ putting more nodes along the column$)$ AND the number of iterations in Power Method, such that the numerically$-$approximated buckling load is within $\backslash (1\backslash \%\backslash )$ from the analytical value. $($i$.$e$.,$ relative true error

Please for the coding parts do it in MATLAB.