(a) For each non-negative integer $n$ and positive constant $\lambda$, let

$$I_{n}(\lambda)=\int_{0}^{\infty} x^{n} e^{-\lambda x} d x$$

By differentiating $I_{n}$ with respect to $\lambda$, find its value in terms of $n$ and $\lambda$.

(b) By making the change of variables $x=u+v, y=u-v$, transform the differential equation

$$\frac{\partial^{2} f}{\partial x \partial y}=1$$

into a differential equation for $g$, where $g(u, v)=f(x, y)$.