Example 1. Form a differential equation for which $y_{1}(x)=e^{x}, y_{2}(x)=e^{-x}$ form a fundamental system of solutions.
Solution. Applying formula (42), we get

$$
\left|\begin{array}{rrr}
e^{x} & e^{-x} & y \\
e^{x} & -e^{-x} & y^{\prime} \\
e^{x} & e^{-x} & y^{\prime \prime}
\end{array}\right|=0, \quad \text { or } \quad\left|\begin{array}{rrr}
1 & 1 & y^{\prime} \\
1 & -1 & y^{\prime} \\
1 & 1 & y^{\prime \prime}
\end{array}\right|=0
$$

Expanding the determinant on the left side of (43) by the elements of the third column, we will have \( y^{\prime \prime} - y = 0 \). This is the required differential equation.