Consider the second order differential equation

$1\mathrm{.}8767$x $($t$)+3\mathrm{.}1245$x $($t$)+0\mathrm{.}6512$x$($t$)=0$

with initial conditions

November $2,2023$

x$(0)=1\mathrm{.}4714,$ x $(0)=3\mathrm{.}4303$

Construct the corresponding matrix A that converts this second order differential equation into a

two$-$dimensional system of first order linear differential equations. Write a Matlab script that

$1.$ Calculates the eigenvalues and eigenvectors and constructs the matrices P$,$ D and P$1,$ that can easily be changed to accept a different differential equation, and

$2.$ Graphs the solution t versus x$($t$).$ Make sure the graph has a title which includes your name and the axis are appropriately labelled. Include a grid.

Print your matlab script and the graph you produced and hand in for grading. Work through the theory first and write an algorithm as to what you need do first.