Science:Math Exam Resources/Courses/MATH215/December 2011/Question 06 (b)
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Question 06 (b) 

Let (b) Find the general solution of the system of equations Here Failed to parse (unknown function "\begin{array}"): {\displaystyle \mathbf{x}(t) = \left(\begin{array}{c} x_1(t) \\ x_2(t) \end{array}\right) } 
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 

If the system is "diagonalized", it can be solved easily with a linear change of variables. 
Hint 2 

How does having a diagonalized matrix M such that help us solve the problem Consider y = S^{1}x. 
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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Eigenvector vKnowing , we will proceed to find our eigenvectors by solving .
which gives the homogeneous system . Placing this in reducedrow form we get so that . We take as the eigenvector. Unfortunately, we only find one eigenvector from this, and we'll need to consider generalized eigenvectors and the Jordan canonical form in order to solve this problem. Find the generalized eigenvector wThe generalized eigenvector satisfies which gives us
Row reductions lead to
which has a solution set that can be parameterized by : For simplicity, we can take so that our generalized eigenvector is
Setting up the Jordan normal form MHaving found the generalized eigenvector, we get that the matrix A has Jordan Canonical form where
A quick computation yields Our system states that where We can therefore write, where
Solve the simplified system y' = MyThe system tells us . The second equation gives us Using this in the first equation means which has been solved using integrating factors. Now we have Recover the solution xwhich immediately gives us our solution x, . 