# Science:Math Exam Resources/Courses/MATH152/April 2022/Question B5 (c)/Solution 1

From the hint, we know that we may find a solution to a differential equation of the form as a function , where is an eigenvalue-eigenvector pair of . A general solution to the differential equation is a linear combination of “eigensolutions” of this type.

If we want a non-oscillating solution, then we will need to be real-valued (see below for an explanation). From the given eigendecomposition of we see that the third eigenvalue is real-valued. Thus, a non-oscillating solution is

To see why must be real-valued, consider an eigenvalue . Then has sine and cosine terms as long as . Thus, to make sure there are no sine and cosine terms we have to take , meaning is real-valued.