Solving a System of Differential Equations by Transformation

Solve the system by transforming it into a single differential equation: $X' = -2Y, Y' = X$.

When solving systems of differential equations, one effective strategy is to simplify the system by transforming it into a single differential equation. This process involves using algebraic manipulations to express one of the variables in terms of derivatives of the other. In this problem, we start with a system where the derivative of X is expressed in terms of Y, and vice versa. This interdependence can be exploited to eliminate one variable and reduce the system to a single differential equation in one of the variables. This often involves differentiating one equation and substituting from the other equation. This method is particularly useful in systems characterized by linear relationships or cycles, like those found in certain mechanical and electrical oscillation problems.

Understanding how to convert a system into a single equation not only aids in simplification but can also enhance insight into the system's behavior. By reducing the problem to a single equation, one can leverage techniques for solving linear differential equations, while also discerning patterns or symmetries that might not be immediately apparent in the original system form. This approach is an important tool in the broader context of analyzing linear systems of differential equations, underpinning methods used in diverse applications such as control theory and dynamic modeling.