Science:Math Exam Resources/Courses/MATH152/April 2012/Question 08 (b)
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q6 (e) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) •
Question 08 (b) 

Consider the 2 x 2 system of differential equations where is the vector of unknown functions. Find the corresponding eigenvectors. 
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 

The eigenvectors are the nontrivial vectors that satisfy Setting the determinant to be zero (when finding the eigenvalues) is what makes such vectors possible. 
Hint 2 

Calculate vectors and which satisfy and These correspond to the two eigenvalues from part (a). 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. We first calculate the eigenvector of the eigenvalue ,
Considering the first line, we obtain Notice that there is a free variable which is common when finding eigenvectors. Therefore, we can choose , then and have the eigenvector .
Considering the first line, we obtain There is a free choice like before and we choose , then and have the eigenvector . 