Given \(\frac{dy}{dx} = \frac{k}{\sqrt{x}}\), integrate to find \(y\):

\(y = \int \frac{k}{\sqrt{x}} \, dx = k \int x^{-1/2} \, dx = k \cdot \frac{x^{1/2}}{1/2} + c = 2k\sqrt{x} + c\).

Substitute the point \(P(1, -1)\):

\(-1 = 2k\sqrt{1} + c \Rightarrow -1 = 2k + c\).

Substitute the point \(Q(4, 4)\):

\(4 = 2k\sqrt{4} + c \Rightarrow 4 = 4k + c\).

Solving the system of equations:

\(-1 = 2k + c\)

\(4 = 4k + c\)

Subtract the first equation from the second:

\(4 - (-1) = 4k - 2k\)

\(5 = 2k \Rightarrow k = \frac{5}{2}\).

Substitute \(k = \frac{5}{2}\) into \(-1 = 2k + c\):

\(-1 = 2 \times \frac{5}{2} + c \Rightarrow -1 = 5 + c \Rightarrow c = -6\).

The equation of the curve is:

\(y = 5\sqrt{x} - 6\).