Science:Math Exam Resources/Courses/MATH152/April 2022/Question B4 (b)/Solution 1

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The solution to a system of differential equations whose eigenvalues form a complex conjugate pair only requires writing out the solution for one of the eigenvalues, since we can quickly obtain the other by complex conjugation. We start with :

The complex conjugate yields the other part of the solution:

Then, to get the general form, we take the real and imaginary parts of the general complex solution :

which we may rewrite as

since the constants are are arbitrary.