Science:Math Exam Resources/Courses/MATH215/December 2011/Question 06 (b)
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) •
Question 06 (b)
(b) Find the general solution of the system of equations
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.
If the system is "diagonalized", it can be solved easily with a linear change of variables.
How does having a diagonalized matrix M such that
help us solve the problem
Consider y = S-1x.
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.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
Knowing , we will proceed to find our eigenvectors by solving .
which gives the homogeneous system
Placing this in reduced-row form we get
so that . We take
as the eigenvector.
Unfortunately, we only find one eigenvector from this, and we'll need to consider generalized eigenvectors and the Jordan canonical form in order to solve this problem.
Find the generalized eigenvector w
The generalized eigenvector satisfies which gives us
Row reductions lead to
which has a solution set that can be parameterized by :
For simplicity, we can take so that our generalized eigenvector is
Setting up the Jordan normal form M
Having found the generalized eigenvector, we get that the matrix A has Jordan Canonical form
A quick computation yields
Our system states that
We can therefore write,
Solve the simplified system y' = My
The second equation gives us
Using this in the first equation means
which has been solved using integrating factors. Now we have
Recover the solution x
which immediately gives us our solution x,