The position $x$ of the leading edge of an avalanche moving down a mountain side making a positive angle $\alpha$ to the horizontal satisfies the equation

$$\frac{d}{d t}\left(x \frac{d x}{d t}\right)=g x \sin \alpha$$

where $g$ is the acceleration due to gravity.

By multiplying the equation by $x \frac{d x}{d t}$, obtain the first integral

$$x^{2} \dot{x}^{2}=\frac{2 g}{3} x^{3} \sin \alpha+c$$

where $c$ is an arbitrary constant of integration and the dot denotes differentiation with respect to time.

Sketch the positive quadrant of the $(x, \dot{x})$ phase plane. Show that all solutions approach the trajectory

$$\dot{x}=\left(\frac{2 g \sin \alpha}{3}\right)^{\frac{1}{2}} x^{\frac{1}{2}}$$

Hence show that, independent of initial conditions, the avalanche ultimately has acceleration $\frac{1}{3} g \sin \alpha$.