Task $1$ In your MATLAB script file, write the code required to find the eigenvalues and

$\{$:${}_2)$ and the corresponding eigenvectors and $x_2)$ for the matrix $A.$ Use the corresponding

section of the 'skeleton' MATLAB script at the end of this assignment. Run your script file to

find ${}_1,{}_2,x_1$ and $x_2.$ Write your answers in your report.

The general solution of the system of equations $(4)$ will be a linear combination of all of the

solutions described above.

Question $4$ Write the general solution in terms of the eigenvalues and eigenvectors of the

matrix $A.$

Hint: The system of equations $(4)$ consists of two second order linear ODEs. This means

that we should expect the general solution to have four arbitrary constants, that is$,$ the general

solution should be a linear combination of four solutions. Use $c_1,c_2,c_3,c_4$ for the arbitrary

constants.

The values of $c_1,c_2,c_3,c_4$ will depend on the initial conditions of the system, that is$,$ the specific

initial value problem. We will consider the following specific initial value problem:

$z_1(0)=1m,z_2(0)=5m$

$z_1^{}(0)=0m{s}^{-1},z_2^{}(0)=0m{s}^{-1}$

Question $5$ Give a physical interpretation of the initial conditions listed above in $(5).$

Question $6$ To determine the unknown constants for any initial value problem, we can solve

a matrix equation of the form $Bc=d$ where $c=[c_1,c_2,c_3,c_4{]}^T.$ Write down $B$ and $d.$

Task $2$ Then, in your MATLAB script file, write the code required to find $c_1,c_2,c_3,c_4$ for the

specific initial value problem described above. Be sure, though, to set it up so that you can easily

change the initial value problem.

Our solution, $Z,$ describes the positions of the point masses at a particular time $t.$

Task $3$ Fill in the necessary MatLaB code required to calculate the positions of the masses at

times $0t40$ seconds $($use a stepsize of $0\mathrm{.}05$s$).$

To create the animation, we use the for loop within the skeleton MATLAB code provided. Copy

this to your MATLAB script file. Run the animation $-$ this will create MATLAB Figure $1.$

Question $7$ Explain precisely what the lines of code marked with $(***)$ do$.$

Task $4$ Create a MATLAB figure that shows the path traced out by each mass as a function of

time $($ie for each mass, plot the position on the $y-$axis and time on the $x-$axis$).$ Show both traces

on the same figure. Only plot the graph for $0t20$ using the axis command. Include a

printout of this figure in your report $($Figure $2).$

Question $8$ Explain why this does not need to be done inside the animation loop.Rest length of springs non$-$zero

If the rest length of the springs is not zero $($i$.$e$.L0),$ then the system of equations $(4)$ is

non$-$homogeneous. In this case, the solution will be given by the sum of the solution to the

homogeneous system and a particular solution:

$Z=Z_h+Z_p$

where $Z_h$ is the solution from Question $4$ and $Z_p$ is given by $Z_p=A^{-1}L.$ For Question $9$ and

Tasks $5-9,$ assume the springs both have rest length $l=2m.$

Task $5$ In your MatLaB script file, add the extra term $(A^{-1}L)$ to your solution for $Z.$

Task $6$ Create a MatLaB figure that plots $z_2(t)($y$-$axis$)$ against $z_1(t)(x-$axis$)$ for $0t40$

$($ie plot the position of the second mass against the position of the first$).$ Be sure to label your

axes and include a printout of this figure in your report $($Figure $3).$ This is an example of a

Lissajous curve. $($You do not need to know this, but you might be interested in the graph that

it creates!$)$

It is also interesting to calculate the velocity of the masses.

Question $9$ By differentiating your general solution, write down an expression for the velocity

of the masses $($write your answer in vector form$).$

Task $7$ Fill in the necessary MATLAB code required to calculate the velocities of the masses at

times $0t40.$

Task $8$ Create a MatLaB figure that plots the velocity of the first mass, $u_1(t)=d\frac{z_1}{d}t($y$-$axis$)$

against