Pursuit Curve of Boat in River

Theorem
Consider a straight river $R$ whose parallel banks are aligned with the $y$-axis and the line $x = c$ of a cartesian plane.

Let the current of $R$ have a constant and uniform speed $a$ in the negative $y$ direction.

Let a boat $B$ be launched from the point $\tuple {c, 0}$ and headed directly towards the origin with speed $b$ relative to the water.

The path of $B$ is defined by the equation:
 * $c^k \paren {y + \sqrt {x^2 + x^2} } = x^{k + 1}$

where $k = \dfrac a b$.

Proof
The components of the velocity of $b$ are:
 * $\dfrac {\d x} {\d t} = - b \cos \theta$
 * $\dfrac {\d y} {\d t} = - a + b \sin \theta$

Hence:

$(1)$ is a homogeneous differential equation.

Let $z = \dfrac y x$.

Then by Solution to Homogeneous Differential Equation:

Substituting back for $z$, putting $k = \dfrac a b$, and rearranging:
 * $C^k \paren {y + \sqrt {x^2 + x^2} } = x^{k + 1}$