Chapter 4 - Vector Spaces - 4.8 Applications of Vector Spaces - 4.8 Exercises - Page 219: 14

$$W=-3 \mathrm{e}^{2 x} \sin (3 x)-2 \cos (3 x) \mathrm{e}^{2 x}.$$

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