Science:Math Exam Resources/Courses/MATH152/April 2022/Question A17/Solution 1

From UBC Wiki

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 .

  • : Let , then

We can see that me must have , and that is a free variable. We choose it to be , so .

  • : Let , then

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: