Integrate \( \frac{dy}{dx} \):

\( y=\int \left(2x-6x^{\frac12}\right)\,dx \).

\( y=x^2-4x^{\frac32}+C \).

Use the point \( (4,-9) \):

\( -9=4^2-4(4^{\frac32})+C \).

\( -9=16-4(8)+C=-16+C \).

So \( C=7 \).

Answer: \( y=x^2-4x^{\frac32}+7 \).