Orthogonal Trajectories/Examples/Circles Tangent to Y Axis/Proof 1

Proof
We use the technique of formation of ordinary differential equation by elimination.

Differentiating $(1)$ $x$ gives:
 * $2 x + 2 y \dfrac {\d y} {\d x} = 2 c$

from which:
 * $\dfrac {\d y} {\d x} = \dfrac {y^2 - x^2} {2 x y}$

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:
 * $\dfrac {\d y} {\d x} = \dfrac {2 x y} {x^2 - y^2}$

Let:
 * $\map M {x, y} = 2 x y$
 * $\map N {x, y} = x^2 - y^2$

Put $t x, t y$ for $x, y$:

Thus both $M$ and $N$ are homogeneous functions of degree $2$.

Thus, by definition, $(1)$ is a homogeneous differential equation.

By Solution to Homogeneous Differential Equation, its solution is:
 * $\displaystyle \ln x = \int \frac {\d z} {\map f {1, z} - z} + C$

where:
 * $\map f {x, y} = \dfrac {2 x y} {x^2 - y^2}$

Thus: