The profile of a cooling fin in a high-performance engine is modeled by a curve $C$ in the $(x,y)$ plane. For $y>0$, the gradient of the curve at any point $(x,y)$ is given by the differential equation

$$\frac{dy}{dx}=\frac{x^3{y}^2}{27}$$

The curve $C$ passes through the point with coordinates $(3,1)$. Show that $C$ intersects the coordinate axes at exactly one point and state the coordinates of this point. Fully justify your answer.