91Ó°ÊÓ

When we are tasked to find the general solution to a system of differential equations like \( \mathbf{x'} = A \mathbf{x} \), it involves finding the eigenvalues and eigenvectors of matrix \( A \). The general solution provides a comprehensive form of how solutions behave over time. Since matrix \( A \) here gives complex eigenvalues \( \lambda = 2 \pm i \), we can expect the solutions to exhibit oscillatory behavior while their amplitude increases.
To find the general complex solution, we first solve the characteristic equation, resulting in complex conjugate eigenvalues indicating oscillation. Next, we find the corresponding complex eigenvectors, here shown as \( \mathbf{v} = \begin{bmatrix} i-1 \ 1 \end{bmatrix} \). The general complex solution takes form \( \mathbf{x}(t) = c_1 e^{(2+i)t} \mathbf{v} \), combining both real and imaginary exponential and trigonometric components.
We express the complex exponential part using Euler's formula \( e^{it} = \cos(t) + i\sin(t) \). Thus, by separating into real and imaginary parts, we extract a solution structure that incorporates both exponential growth and rotational aspects, fundamental to systems characterized by complex eigenvalues.

Once the general complex solution is found, it is crucial to convert it into a real solution, especially when dealing extensively with real-world phenomena. To do this, we separate the complex expression of the solution into two distinct real parts by utilizing the terms from the identities involving sine and cosine.
The real solution of the system is formed by taking the real and imaginary parts of the generalized complex solution:

• The first component \( \mathbf{x}_1(t) = e^{2t} \left( \cos(t) \begin{bmatrix} 1 \ 0 \end{bmatrix} - \sin(t) \begin{bmatrix} 0 \ 1 \end{bmatrix} \right) \) captures cosine-related behavior.
• The second component \( \mathbf{x}_2(t) = e^{2t} \left( \sin(t) \begin{bmatrix} 1 \ 0 \end{bmatrix} + \cos(t) \begin{bmatrix} 0 \ 1 \end{bmatrix} \right) \) reflects sine-oriented effects.

The combination of these components leads to the general real solution \( \mathbf{x}(t) = c_1 \mathbf{x}_1(t) + c_2 \mathbf{x}_2(t) \). It reveals both oscillatory characteristics from \( \sin \) and \( \cos \) terms, alongside the exponential factor \( e^{2t} \) dictating radial expansion over time.

The trajectories of a system like \( \mathbf{x'} = A\mathbf{x} \) depict the paths followed by the system's state over time. In this context, with eigenvalues \( \lambda = 2 \pm i \), we observe that trajectories are spiral outward, showcasing both growth and rotational dynamics.
The exponential term \( e^{2t} \) from the solution increases the amplitude of the state vector, causing the trajectories to expand outward from the origin as time progresses. This means that, without any restricting forces, trajectories move away from the equilibrium at an increasing rate.
Simultaneously, the sine and cosine components impart a rotational movement, ensuring that the state rotates around the origin, creating a spiraling effect. The real part of the eigenvalues, \( 2 \), confirms that these spirals are not merely stationary but expand progressively, without oscillations dying out. Thus, these trajectories signify a dynamic shift involving both directional rotation and radial growth out from the starting point.