MATH152 April 2022
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Question A17
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Let
Find the general solution to the system of differential equations . (Note: this problem requires that you first find the eigenvalues and eigenvectors of the matrix.)
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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?
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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!
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Hint
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Recall that the form for the general solution to a differential equation system is .
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Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
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Solution
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Recall from the hint that the general solution to a differential equation of this form is , so to to determine the general solution to the given system, we need to compute the eigenvalues and eigenvectors of .
Since is an upper triangular matrix, we can read the eigenvalues off the diagonal, they are: and . To find the corresponding eigenvectors, we will use the fact that .
- : Let , then
We can see that me must have , and that is a free variable. We choose it to be , so .
- : Let , then
The first equation simplifies to , and the second equation imposes no restrictions on what has to be, so we can choose , which results in and .
Now, we can write the general solution to the system of differential equations:
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