4. (1.5 points) Which of the following describes the equilibrium point of the following system of differential equations: \[ \frac{d Y}{d t}=\left(\begin{array}{cc}10 & 3 \\ 2 & 9\end{array}\right) Y \text {. } \] (a) center (b) sink (c) source (d) saddle (e) spiral sink (f) spiral source

To find the equilibrium point of the system of differential equations, we need to analyze the matrix involved. The given matrix is \(\begin{pmatrix} 10 & 3 \\ 2 & 9 \end{pmatrix}\). The first step is to calculate the eigenvalues, which will help determine the stability of the equilibrium point. The characteristic equation is found by solving the determinant \(\text{det}(A - \lambda I) = 0\), where \(A\) is the matrix and \(\lambda\) represents the eigenvalues. This results in a positive real eigenvalue, leading to an understanding that the system behaves as a source due to the properties of the evolution. Thus, the equilibrium point acts like a source. So, the answer is: (c) source. Now, let's dive a little deeper! In systems of differential equations, the nature of equilibrium points can drastically affect the behavior of those systems over time. For instance, if you were to visualize the trajectories of a source, you'd see trajectories diverging outward, representing instability in that equilibrium. Understanding the stability of these points is not just an academic exercise; it has real-world implications in fields like ecology, where populations can grow rapidly under certain conditions, leading to unsustainable scenarios!