Solve a System of First Order Differential Equations

Solve the system by transforming it into a single differential equation: $X' = Y, Y' = 6X - Y$.

Solving systems of differential equations is a fundamental aspect of studying dynamic systems in fields such as physics and engineering. This problem involves transforming a given system of first-order differential equations into a single equation, which promotes a deeper understanding of the interdependencies between different variables in a dynamic system. By focusing on transforming the system, we explore the relationship between the equations, and how one variable rate of change may directly and indirectly affect another.

One effective strategy when working with a system of differential equations is to express one variable in terms of the other. By substituting expressions and simplifying, we can reduce the dimensionality of the problem to a single differential equation, making it easier to solve using techniques applicable to single-variable differential equations. Such an approach not only highlights the underlying link between the equations but also aligns with the methods used to tackle more complex systems.

This procedure is widely applicable in contexts where systems are initially presented with multiple interdependent parts, such as coupled systems found in electrical circuits and mechanical systems. Understanding how to reduce and solve these systems equips one with valuable skills for tackling similar problems, thereby reinforcing the mathematical foundation required for engineering and the physical sciences.