Here we give a simple example of the use of the Mellin transform to solve a differential equation.
Consider the differential equation $\begin{equation*} f'(t) + f(t) = 0.\tag{1} \end{equation*}$ On taking the Mellin transform we find that the corresponding complex difference equation is $-(s-1)\phi(s-1) + \phi(s) = 0$, or $\phi(s+1) = s \phi(s).$ This is the recurrence relation for the gamma function, so $\phi(s) = \Gamma(s)$ up to an overall numerical factor.
The solution to (1) must be $\begin{equation*} f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} ds\, t^{-s} \Gamma(s),\tag{2} \end{equation*}$ where $c$ picks out some suitable contour avoiding the poles of $\Gamma$. Recall that $\Gamma$ is meromorphic with simple poles at $0, -1, -2, \ldots$. Existence of the inverse transform (2) can be shown by appealing to the asymptotic behavior of $\Gamma$, $|\Gamma(x+i y)| \sim \sqrt{2\pi} |y|^{x-1/2} e^{-|y|\pi/2},$ in the limit $|y|\to \infty$.
We choose $c>0$. With this choice of contour the integral (2) is the Cahen-Mellin integral. Pushing the contour to the left, we pick up all the poles of the gamma function. To calculate the residue, recall that near $s=-n$ $\Gamma(s) = \frac{(-1)^n}{n!}\frac{1}{s+n} + O(1).$ This is a straightforward consequence of the recurrence relation for $\Gamma$. Thus, $\begin{eqnarray*} f(t) &=& \sum_{n=0}^\infty \mathrm{Res}_{s=-n} t^{-s} \Gamma(s) \\ &=& \sum_{n=0}^\infty t^{n}\frac{(-1)^n}{n!} \\ &=& e^{-t}. \end{eqnarray*}$