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Task 1 In your Matlab script file, write the code required to find the eigenvalues ( lambda 1 and lambda 2 ) and

Task 1 In your Matlab script file, write the code required to find the eigenvalues (\lambda 1 and
\lambda 2) and the corresponding eigenvectors (x1 and x2) 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 \lambda 1,\lambda 2, x1 and x2. 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 c1, c2, c3, c4 for the arbitrary
constants.
The values of c1, c2, c3, c4 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:
z1(0)=1 m z2(0)=5 m
z1(0)=0 m s1
z2(0)=0 m s1
(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 =[c1, c2, c3, c4]
T
. Write down B and d.
Task 2 Then, in your Matlab script file, write the code required to find c1, c2, c3, c4 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 0.05s).
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.

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