91Ó°ÊÓ

When dealing with complex eigenvalues in a differential equation, it's essential to express solutions in terms of real numbers. Starting with complex numbers can be intimidating, but rest assured, it's a standard procedure to transform these into real solutions. In the exercise, we faced eigenvalues of \( \lambda = -3 \pm i \). These lead to exponential expressions involving both sine and cosine functions after using Euler's formula. This way, we construct real solutions, which can be more intuitive and easier to interpret in the real world. The key takeaway is understanding that even with complex numbers, we aim to express our results in real forms.

Eigenvectors play a critical role when it comes to finding solutions for a system of differential equations. Once eigenvalues are found, eigenvectors help to shape the particular solutions of the differential equation system. In our case, for the eigenvalues \( \lambda = -3 + i \) and \( \lambda = -3 - i \), we derived complex eigenvectors. These vectors are inherently tied to the system of equations we're solving, and their real and imaginary parts directly contribute to forming the basis of the solution space. Recognizing the importance of choosing a suitable eigenvector is key to accurately describing the differential system.

The general solution to a differential equation system uses both the eigenvector and eigenvalue information derived earlier. Combining these insights, the exercise's general solution involved forming two real-valued functions from the complex eigen pair. By applying trigonometric identities and combining real and imaginary parts of eigenvectors, we rendered pairs of sinusoidal functions multiplied by exponential decays, specifically: \( \mathbf{y}_1 = e^{-3t} \begin{pmatrix} \cos t \ -2\cos t + \sin t \end{pmatrix} \) and \( \mathbf{y}_2 = e^{-3t} \begin{pmatrix} \sin t \ -2\sin t - \cos t \end{pmatrix} \). These represent the fundamental real solutions satisfying the system, and together, they characterize the comprehensive behavior of the system over time.

Verifying linear independence is the final key step when confirming our solutions are distinct and essential. If solutions are not linearly independent, they can't span the solution space properly, leading to incomplete results. In our context, using the Wronskian determinant ensures this independence. Here, the computed Wronskian, \(W = e^{-6t}\), stayed non-zero, validating that our constructed solutions, \( \mathbf{y}_1 \) and \( \mathbf{y}_2 \), do indeed form a valid basis for the solution space. This means they don't overlap in a way that would cause them to be redundant, hence capturing all potential solution behaviors.