Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 27
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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! 
Hint 

Compute the eigenvalues and eigenvectors of the coefficient matrix If has two eigenvalues with associated eigenvectors , then for some constants . 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 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 