The stability of eigenvalues and eigenvectors and their impact on differential systems

In this article, we apply the stability of eigenvalues and eigenvectors and their impact on differential systems. To achieve this goal, the eigenvalues and eigenvectors are studied and their differential systems, nature in terms of being different real values, compound eigenvalues, or equal eigenvalues.And to identify how to solve linear differential systems with fixed coefficients with the initial condition a complete solution, which depends on the eigenvalues and the corresponding eigenvectors, finding the general solution and the geometry of teigenvectors graphically and the effect of theigenvalues for the three eigenvalues cases, by drawing the  paths and the phase plane and clarifying the state of equilibrium contract and stability