Use the Eigenvalue Method to solve the following system of ODES

Question

Use the Eigenvalue Method to solve the following system of ODES:

 

 

Hint: The corresponding coefficient matrix has a pair of complex eigenvalues.

Answer 1

Well her goes: You subtract the eigenvalues on the main diagonal and you get the equation for λ by zeroing out the determinant: (-2-λ)² + 4 = 0 → λ = -2 ± -2i

Plugging the eigenvalue, -2 + 2i, back into the matrix, you get row 1 as -2i 1 and row 2 as -4 2i. This implies an eigenvector, ξ, of (1 2i)^T

Since we have a complex eigenvalue we will use the form (ξ)e^-2t(cos(2t) + isin(2t))

Separating the real and imaginary parts, we obtain:

e^-2t | cos(2t) | + e^-2t | sin(2t) | = |x|

| -2sin(2t)| | 2cos(2t)| |y|

I hope that helps. I checked X', but not Y' equation.