"[Here] we explore a relationship between determinants and solutions to a differential equation. The $3\times 3$ matrix consisting of solutions to a differential equation and their derivatives is called the Wronskian and, as we will see in later chapters, plays a pivotal role in the theory of differential equations."

This is the question following the description:

"Verify that ${y}_{1}(x)=\mathrm{cos}(2x),{y}_{2}(x)=\mathrm{sin}(2x),{y}_{3}(x)={e}^{x}$ are solutions to the differential equation: ${y}^{\u2034}-{y}^{\u2033}+4{y}^{\prime}-4y=0$, and show that $\{\{{y}_{1},{y}_{2},{y}_{3}\},\{{y}_{1}^{\prime},{y}_{2}^{\prime},{y}_{3}^{\prime}\},\{{y}_{1}^{\u2033},{y}_{2}^{\u2033},{y}_{3}^{\u2033}\}\}$ is nonzero on any interval."

Now I'm really just looking for an explanation on how I would show that it is nonzero on any interval. I completed the first part, I believe, by just taking up to the third derivative for each and plugging each set in to verify an identity of 0=0 at the end. I would then put the terms into the matrix which would give me:

$$\{\{\mathrm{cos}(2x),\mathrm{sin}(2x),{e}^{x}\},\{-2\mathrm{sin}(2x),2\mathrm{cos}(2x),{e}^{x}\},\{-4\mathrm{cos}(2x),-4\mathrm{sin}(2x),{e}^{x}\}\}$$

So I've gotten this far and if anyone could point me in the right direction on where to go from here it would be greatly appreciated.