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

Integrate: \(y = \int (2x + 1)^{\frac{1}{2}} \, dx\).

Let \(u = 2x + 1\), then \(du = 2 \, dx\) or \(dx = \frac{1}{2} \, du\).

Substitute: \(y = \int u^{\frac{1}{2}} \cdot \frac{1}{2} \, du = \frac{1}{2} \int u^{\frac{1}{2}} \, du\).

Integrate: \(\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} = \frac{1}{3} (2x + 1)^{\frac{3}{2}} + c\).

Thus, \(y = \frac{(2x + 1)^{\frac{3}{2}}}{3} + c\).

Use the point \((4, 7)\) to find \(c\):

\(7 = \frac{(2 \cdot 4 + 1)^{\frac{3}{2}}}{3} + c\).

\(7 = \frac{9^{\frac{3}{2}}}{3} + c\).

\(7 = \frac{27}{3} + c\).

\(7 = 9 + c\).

\(c = -2\).

Therefore, the equation of the curve is \(y = \frac{(2x + 1)^{\frac{3}{2}}}{3} - 2\).