Determining Linear Independence Using Wronskian

For two functions $y_1 = e^{2x}$ and $y_2 = e^{-2x}$, determine if they are linearly independent using the Wronskian test.

In this problem, you are asked to determine whether two functions are linearly independent by using the Wronskian test. The Wronskian is a determinant used in the theory of differential equations to help ascertain linear independence of a set of functions. For two functions in particular, the Wronskian involves taking the determinant of a 2x2 matrix composed of those functions and their derivatives. If the Wronskian is non-zero at some point within an interval, the functions are linearly independent on that interval. Conversely, if the Wronskian is zero everywhere on the interval, the functions may be linearly dependent; however, a zero Wronskian does not always guarantee linear dependence, making it a useful but not definitive test.

The concept of linear independence is fundamental in linear algebra and plays a crucial role in solving differential equations, particularly in verifying the uniqueness of solutions. In this problem, the specific functions given are exponential functions, making it a classic case when using the Wronskian test. Evaluating the Wronskian here will involve computing derivatives of exponential functions, which are notably straightforward due to their property of having derivatives proportional to themselves. This problem illustrates how the Wronskian is used practically and emphasizes the interplay between linear algebra concepts and differential equation theory.