To find the equation of the curve, we need to integrate the gradient function \(\frac{dy}{dx} = x - 3x^{-\frac{1}{2}}\).

Integrate term by term:

\(\int x \, dx = \frac{x^2}{2}\)

\(\int -3x^{-\frac{1}{2}} \, dx = -3 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -6x^{\frac{1}{2}}\)

Thus, the integral is:

\(y = \frac{x^2}{2} - 6x^{\frac{1}{2}} + c\)

We use the point \((4, 1)\) to find \(c\):

\(1 = \frac{4^2}{2} - 6 \cdot 4^{\frac{1}{2}} + c\)

\(1 = 8 - 12 + c\)

\(1 = -4 + c\)

\(c = 5\)

Therefore, the equation of the curve is:

\(y = \frac{1}{2}x^2 - 6x^{\frac{1}{2}} + 5\)