Consider the differential equation

$$x y^{\prime \prime}+(a-x) y^{\prime}-b y=0$$

where $a$ and $b$ are constants with $\operatorname{Re}(b)>0$ and $\operatorname{Re}(a-b)>0$. Laplace's method for finding solutions involves writing

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

for some suitable contour $C$ and some suitable function $f(t)$. Determine $f(t)$ for the equation $(*)$ and use a clearly labelled diagram to specify contours $C$ giving two independent solutions when $x$ is real in each of the cases $x>0$ and $x<0$.

Now let $a=3$ and $b=1$. Find explicit expressions for two independent solutions to $(*)$. Find, in addition, a solution $y(x)$ with $y(0)=1$.