.

Then $D_x u_1 = \frac{-y_2 g}{w(y_1, y_2)}$ and $D_x u_2 = \frac{y_1 g}{w(y_1, y_2)}$. This tells us that $y_p = u_1 y_1 + u_2 y_2$ (notice that $u_1, u_2$ are not derivatives).