Chapter 4 - Vector Spaces - 4.8 Applications of Vector Spaces - 4.8 Exercises - Page 219: 20

$$W=-2 x \mathrm{e}^{x} \mathrm{e}^{-x}.$$

The Wronskian of $\left\{x, e^{-x}, e^{x}\right\}$ is given by $$W=\left|\begin{array}{ccc}{x} & {\mathrm{e}^{-x}} & {\mathrm{e}^{x}} \\ {1} & {-\mathrm{e}^{-x}} & {\mathrm{e}^{x}} \\ {0} & {\mathrm{e}^{-x}} & {\mathrm{e}^{x}}\end{array}\right|=-2 x \mathrm{e}^{x} \mathrm{e}^{-x}.$$