Consider the differential equation

$$z \frac{d^{2} y}{d z^{2}}-2 \frac{d y}{d z}+z y=0$$

Laplace's method finds a solution of this differential equation by writing $y(z)$ in the form

$$y(z)=\int_{C} e^{z t} f(t) d t$$

where $C$ is a closed contour.

Determine $f(t)$. Hence find two linearly independent real solutions of $(\star)$ for $z$ real.