Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 06 (c)
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Question B 06 (c) 

Consider the threecomponent differential equation . The matrix has real entries. It has an eigenvalue and an eigenvalue with corresponding eigenvectors and (c) Describe all initial conditions for which the solution exhibits oscillatory behaviour. Justify your answer briefly. 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

A solution with oscillatory behavior will have a term involving or . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall the general solution as in part (b):
If a solution doesn't have oscillatory behaviour, it means that a solution doesn't contain a term having and . In our case, that happens when . Now, we will relate this condition with the initial data and exclude this case. Suppose the initial condition is Setting in the general solution as in part (b), we get and Then, we can easily see that is equivalent with , where is an arbitrary number. Therefore, when , the corresponding initial data doesn't have oscillatory behaviour. In other words, the condition on initial data to make the corresponding solution exhibit oscillatory behavior is that 