= Solution
For a <point vortex in stagnation flow beside a wall>, a freely moving <point vortex> is transported by the regular <velocity> due to the background and its image, excluding its own singular <velocity>. Differentiate the potential and evaluate that regular part at $z_0=x+iy$:
$$
\dot x-i\dot y=z_0+\frac{i\Gamma}{2\pi(z_0-\overline z_0)}
=x+iy+\frac{\Gamma}{4\pi y}.
$$
Thus the equations of motion are
$$
\boxed{\dot x=x+\frac{\Gamma}{4\pi y},\qquad\dot y=-y.}
$$
Writing $c=\Gamma/(4\pi)$, differentiate the proposed path function:
$$
\frac d{dt}(xy+c\log y)=\left(x+\frac cy\right)y+x(-y)+c\frac{-y}{y}=0.
$$
Hence
$$
\boxed{xy+\frac{\Gamma}{4\pi}\log y=\text{constant}.}
$$
For initial position $(x_0,y_0)$, an explicit check is $y(t)=y_0e^{-t}$ and $x(t)=e^t(x_0+ct/y_0)$. Their substitution makes the linear terms in $t$ cancel in the path function and also shows that the vortex stays above the wall at every finite time.
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