Science:Math Exam Resources/Courses/MATH152/April 2022/Question B5 (c)
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Question B5 (c) 

With the variables ordered , let the matrix of the system of differential equations be . (You don’t need to find .) Consider the following Matlab result: > [P,D] = eig(A) P = 0.5  0.34i 0.5 + 0.34i 0.22 + 0.00i 0.62 + 0.00i 0.620.00i 0.85 + 0.00i 0.43  0.21i 0.43 + 0.21i 0.47 + 0.00i D = Diagonal Matrix 0.15 + 0.17i 0 0 0 0.15  0.17i 0 0 0 0.77 + 0.00i There exist initial conditions that result in a nonoscillating solution (a real solution that does not involve sin and cos). Find one such solution , , that is not oscillating and not zero. Note: you don’t need the results from (a) and (b) to attempt (c). 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Recall, again, that solutions to a differential equation of the form have the form (the general solution is the sum of multiple of these solutions, one for each eigenvalue). If we want a real valued solution that doesn’t involve sine and cosine, what does this mean for the eigenvalues and eigenvectors? 
Hint 2 

The matlab command yields the matrix , whose columns are the eigenvectors of , and the diagonal matrix , whose entries are the eigenvalues of . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. From the hint, we know that we may find a solution to a differential equation of the form as a function , where is an eigenvalueeigenvector pair of . A general solution to the differential equation is a linear combination of “eigensolutions” of this type. If we want a nonoscillating solution, then we will need to be realvalued (see below for an explanation). From the given eigendecomposition of we see that the third eigenvalue is realvalued. Thus, a nonoscillating solution is
To see why must be realvalued, consider an eigenvalue . Then has sine and cosine terms as long as . Thus, to make sure there are no sine and cosine terms we have to take , meaning is realvalued. 