Chapter 4 - Vector Spaces - 4.8 Applications of Vector Spaces - 4.8 Exercises - Page 219: 18

$$W=x.$$

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