Example 7. Find the Wronskian determinant for the functions $y_{1}(x)=e^{k_{1} x}$, $y_{2}(x)=e^{k_{2} x}, y_{3}(x)=e^{k_{3} x}$.

1) The linear dependence of the functions $\sin x, \sin \left(x+\frac{\pi}{8}\right), \sin \left(x-\frac{\pi}{8}\right)$ can be established by noting that $\sin \left(x+\frac{\pi}{8}\right)+\sin \left(x-\frac{\pi}{8}\right)=2 \cos \frac{\pi}{8} \sin x$, or $\sin \left(x+\frac{\pi}{8}\right)+$ $\sin \left(x-\frac{\pi}{8}\right)-2 \cos \frac{\pi}{8} \sin x \equiv 0$.
## Solution. We have

$W\left[y_{1}, y_{2}, y_{3}\right]=\left|\begin{array}{lll}e^{k_{1} x} & e^{k_{2} x} & e^{k_{3} x} \\ k_{1} e^{k_{1} x} & k_{2} e^{k_{2} x} & k_{3} k_{1} k_{3} \\ k_{1}^{2} e^{k_{1} x} & k_{2}^{2} e^{k_{2} z} & k_{3}^{2} e^{k_{3} x}\end{array}\right|=e^{\left(k_{1}+k_{2}+k_{3}\right) x}\left(k_{2}-k_{1}\right)\left(k_{3}-k_{1}\right)\left(k_{3}-k_{2}\right) . \Delta$