Eigenvalues and Eigenvectors of a System of Differential Equations

Given a system of two first-order, linear, homogeneous differential equations with distinct real eigenvalues, determine the eigenvalues and eigenvectors, and write down the general solution.

When dealing with a system of first-order linear homogeneous differential equations, a powerful technique is to find the eigenvalues and eigenvectors of the system's coefficient matrix. This involves solving the characteristic equation, which is derived from the determinant of the matrix minus a scalar multiple of the identity matrix. The roots of this characteristic equation are the eigenvalues. Each eigenvalue will correspond to an eigenvector, which can be found by solving the system of equations that arises when substituting the eigenvalues back into the matrix equation.

Understanding the geometric and algebraic multiplicity of these eigenvalues is crucial. For systems with distinct real eigenvalues, as in this problem, the solution can often be expressed as a linear combination of the eigenvectors multiplied by exponential functions of the eigenvalues and the independent variable. This method leverages the structure of the differential equation system, simplifying the task of finding a general solution.

Embracing these concepts allows one to translate the abstract mathematical framework into concrete solutions, exhibiting how systems of equations can be broken down using linear algebra techniques. This approach not only finds applications in mathematics but also provides insights into various physical systems in engineering and sciences, such as electrical circuits and mechanical systems, where such differential equations commonly arise.