 Solution

The divergence is ${\partial }_x{u}_x+{\partial }_y{u}_y=1-1=0$, while the scalar vorticity is ${\partial }_x{u}_y-{\partial }_y{u}_x=0$. The flow is therefore incompressible flow and irrotational flow. Integrating the streamfunction equations givesThe time dependence must be retained: no time-independent function $\psi (x,y)$ represents this unsteady field for every $t$. At $t=0$ the streamlines are $xy=\mathrm{constant}$, with the coordinate axes as separatrices. Their arrows point towards increasing $|x|$ and decreasing $|y|$, as shown below.

A pathline solves the material ODEs, givingFor $x_0\ne 0$, elimination of time yieldsIf $x_0=0$, the pathline stays on the $y$ axis and the parametric expression remains valid; it cannot in general be written as a single-valued $y=f(x)$ along that vertical trajectory. For $x_0=y_0=1$, the path is $y=\log x-1+2/x$: it initially descends, reaches its minimum $(2,\log 2)$, then rises. Unlike a frozen streamline, it records the time-varying flow.