Putnam 1975 A5

Let $I\subset \mathbb{R}$ be an interval and $f(x)$ a continuous real-valued function on $I$ . Let $y_1$ and $y_2$ be linearly independent solutions of $y''=f(x)y$ taking positive values on $I$ . Show that for some positive number $k$ the function $k\cdot\sqrt{y_1 y_2}$ is a solution of $y''+\frac{1}{y^{3}}=f(x)y$ .