Given the function $f(x, y) = e^{ax + \beta y}$, with $a$ $\beta$ constants. Calculate $\mathcal{L}f$, where $\mathcal{L}$ is a differential operator defined as:
a) $\mathcal{L} = x\left(\frac{\partial}{\partial x}\right) + \left(\frac{\partial}{\partial y}\right)y$
b) $\mathcal{L} = \frac{\partial}{\partial y}\left(y + \frac{\partial}{\partial x}\right)$
c) $\mathcal{L} = \frac{\partial}{\partial x}\left(y + \frac{\partial}{\partial x}\right)$
d) $\mathcal{L} = \left(y + \frac{\partial}{\partial x}\right)\frac{\partial}{\partial x}$
e) $\mathcal{L} = \left(\frac{\partial}{\partial x} + \frac{\partial}{\partial y}\right)^2$
