Science:Math Exam Resources/Courses/MATH215/April 2014/Question 01 (e)
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q2 • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) •
Question 01 (e) 

Find the general solution of the differential equation system:

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 

How do the eigenvalues and corresponding eigenvectors of the matrix help to solve the problem? 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. To find the general solution for this system, we need to determine the eigenvalues and corresponding eigenvectors for
We find the eigenvalues by solving the characteristic equation, . This gives the equation , and so and are the eigenvalues for . To find the eigenvector corresponding to , we solve for , i.e. we solve
which implies Analogously for the second eigenvector we solve , i.e. we solve
which yields . The solutions and form a fundamental set of solutions for the system, and therefore the general solution is an arbitrary linear combination of and
where are arbitrary real constants. 