Exponential solutions of a differential equation and a volume of revolution

Consider the differential equation $2\frac{d^2y}{dx^2}-\frac{dy}{dx}-y=0$.

Two solutions of the differential equation have the form $y=e^{mx}$, where $m \in \R$.

Show that $m=-\frac{1}{2}$ or $m=1$.

The graphs of the two solutions of the differential equation are shown on the Cartesian plane below.

The shaded region shown above is bounded by the two curves and the line with equation $x = 1$. This region is rotated about the $x$-axis to form a solid of revolution.

Calculate the volume of this solid.