MATH152 April 2015
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Question A 27
Find the solution of the system of differential equations
that satisfies the initial conditions and .
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!
Compute the eigenvalues and eigenvectors of the coefficient matrix
If has two eigenvalues with associated eigenvectors , then
for some constants .
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To solve this system of differential equations, we begin by finding the eigenvalues of the coefficient matrix
These can be computed by finding the roots of
This quadratic factors into , so the eigenvalues of the coefficient matrix are .
We now compute eigenvectors for each eigenvalue. The eigenvectors of eigenvalue 3 solve the equation
Writing , these equations imply that , and hence . Thus is an eigenvector of eigenvalue 3.
Similarly, for the eigenvalue 1, we have the system
so . Hence is an eigenvector of eigenvalue 1.
The general form of the solution is therefore
Setting and using the initial conditions, we obtain the equations
whence . Thus the solution is