Chapter 4 - Vector Spaces - 4.8 Applications of Vector Spaces - 4.8 Exercises - Page 219: 16

$$W=-4x \mathrm{e}^{x^{2}} \mathrm{e}^{-x^{2}}$$

The Wronskian of $\left\{e^{x^{2}}, e^{-x^{2}}\right\}$ is given by $$W=\left|\begin{array}{cc}{e^{x^{2}}} & {e^{-x^{2}}} \\ {2 x e^{x^{2}}} & {-2 x e^{-x^{2}}}\end{array}\right|=-4x \mathrm{e}^{x^{2}} \mathrm{e}^{-x^{2}}$$