Task $1$ In your Matlab script file, write the code required to find the eigenvalues $(\backslash$lambda $1$ and

$\backslash$lambda $2)$ and the corresponding eigenvectors $($x$1$ 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 $\backslash$lambda $1,\backslash$lambda $2,$ 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$1(0)=1$ m z$2(0)=5$ m

z$1(0)=0$ m s$1$

z$2(0)=0$ m s$1$

$(5)$

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 $0<=$ t $<=40$ 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 $0<=$ t $<=20$ 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.