Given \(\frac{dy}{dx} = \frac{2}{\sqrt{x}} - 1\), we need to find \(y\) by integrating.

Integrate \(\frac{dy}{dx} = \frac{2}{\sqrt{x}} - 1\):

\(y = \int \left( \frac{2}{\sqrt{x}} - 1 \right) \, dx\)

\(y = \int 2x^{-1/2} \, dx - \int 1 \, dx\)

\(y = 2 \cdot \frac{x^{1/2}}{1/2} - x + c\)

\(y = 4\sqrt{x} - x + c\)

Use the point \(P(9, 5)\) to find \(c\):

\(5 = 4\sqrt{9} - 9 + c\)

\(5 = 12 - 9 + c\)

\(5 = 3 + c\)

\(c = 2\)

Thus, the equation of the curve is \(y = 4\sqrt{x} - x + 2\).