Chapter 4 - Vector Spaces - 4.8 Applications of Vector Spaces - 4.8 Exercises - Page 219: 19

The Wronskian of $\left\{e^{-x}, x e^{-x},(x+3) e^{-x}\right\}$ is given by $$W=0.$$

The Wronskian of $\left\{e^{-x}, x e^{-x},(x+3) e^{-x}\right\}$ is given by $$W=\left|\begin{array}{ccc}{\mathrm{e}^{-x}} & {x \mathrm{e}^{-x}} & {(x+3) \mathrm{e}^{-x}} \\ {-\mathrm{e}^{-x}} & {\mathrm{e}^{-x}-x \mathrm{e}^{-x}} & {\mathrm{e}^{-x}-(x+3) \mathrm{e}^{-x}} \\ {\mathrm{e}^{-x}} & {-2 \mathrm{e}^{-x}+x \mathrm{e}^{-x}}&{-2 \mathrm{e}^{-x}+(x+3) \mathrm{e}^{-x}}\end{array}\right|=0.$$