Science:Math Exam Resources/Courses/MATH152/April 2022/Question A17
• QA01 • QA02 • QA03 • QA04 • QA05 • QA06 • QA07 • QA08 • QA09 • QA10 • QA11 • QA12 • QA13 • QA14 • QA15 • QA16 • QA17 • QA18 • QA19 • QA20 • QB1 (a) • QB1 (b) • QB2 (a) • QB2 (b) • QB2 (c) • QB3 (a) • QB3 (b) • QB4 (a) • QB4 (b) • QB4 (c) • QB5 (a) • QB5 (b) • QB5 (c) •
Question A17 

Let Find the general solution to the system of differential equations . (Note: this problem requires that you first find the eigenvalues and eigenvectors of the matrix.) 
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 

Recall that the form for the general solution to a differential equation system is . 
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. Recall from the hint that the general solution to a differential equation of this form is , so to to determine the general solution to the given system, we need to compute the eigenvalues and eigenvectors of . Since is an upper triangular matrix, we can read the eigenvalues off the diagonal, they are: and . To find the corresponding eigenvectors, we will use the fact that .
We can see that me must have , and that is a free variable. We choose it to be , so .
The first equation simplifies to , and the second equation imposes no restrictions on what has to be, so we can choose , which results in and . Now, we can write the general solution to the system of differential equations: 