91Ó°ÊÓ

When dealing with a system of differential equations, eigenvalues and eigenvectors play a crucial role in determining the behavior of the system. To find them, we first need the matrix form of the system. In our problem, the matrix is given by:

• \(A = \begin{bmatrix} 4 & 5 \ -2 & 6 \end{bmatrix}\)

To find the eigenvalues, we solve the equation \(\det(A - \lambda I) = 0\). This involves finding the determinant after substituting the identity matrix scaled by \(\lambda\). For our example:

• Calculate \(\det\begin{bmatrix} 4 - \lambda & 5 \ -2 & 6 - \lambda \end{bmatrix}\)
• Solve \(\lambda^2 - 10\lambda + 34 = 0\) using the quadratic formula, leading to complex roots.

These roots \(\lambda_1 = 5 + 3i\) and \(\lambda_2 = 5 - 3i\) are the eigenvalues. For each eigenvalue, there is a corresponding eigenvector obtained by solving \((A - \lambda I)\mathbf{v} = \mathbf{0}\). The eigenvectors provide the direction of the principal components of the system.

Transforming a system of differential equations into a matrix equation helps simplify and systematically solve the problem. In our system:

• \(\frac{d}{dt}\begin{bmatrix}x \ y\end{bmatrix} = A\begin{bmatrix}x \ y\end{bmatrix}\)

Where \(A\) is the coefficient matrix:

• \(A = \begin{bmatrix} 4 & 5 \ -2 & 6 \end{bmatrix}\)

This matrix equation \(\frac{d\mathbf{X}}{dt} = A\mathbf{X}\) presents a compact way to express the system. It is crucial for finding

• Eigenvalues (\(\lambda\))
• Eigenvectors (\(\mathbf{v}\))

These quantities then guide the construction of the solution. The matrix form reduces solving differential equations to a more manageable algebraic task, centering around linear algebra concepts.

When the characteristic equation yields complex solutions, it indicates oscillatory behavior within the system. For our problem, the characteristic equation yields the complex roots:

• \(\lambda_1 = 5 + 3i\)
• \(\lambda_2 = 5 - 3i\)

Complex solutions typically suggest that the system's solutions have both exponential growth or decay (based on the real part) and oscillatory components (based on the imaginary part). Thus, in a solution:

• The real part (5) indicates exponential growth, expressed as \(e^{5t}\).
• The imaginary part (\(3i\)) brings the trigonometric functions \(\cos(3t)\) and \(\sin(3t)\) to the solution.

Such solutions thus reflect systems where positions or states oscillate over time, typically represented in terms of sine and cosine functions multiplied by an exponential term.

The general solution of a system of differential equations consisting of complex eigenvalues is formed by combining the oscillatory components imparted by these eigenvalues. For our system:

• The eigenvectors create linear combinations based on trigonometric functions.
• Each component in the solution needs both \(\sin(3t)\) and \(\cos(3t)\) terms.
• The exponential term \(e^{5t}\) modulates both trigonometric components.

The general solution, therefore, combines these parts as:

• Real parts of eigenvectors with \(\cos(3t)\) and \(\sin(3t)\)
• Imaginary parts with the opposite trigonometric functions to maintain orthogonality

This results in a comprehensive expression that describes the motion or behavior of the system over time, using coefficients \(c_1\) and \(c_2\) determined by specific initial conditions.