Orthogonal Trajectories/Examples/x + C exp -x

Theorem
Consider the one-parameter family of curves:
 * $(1): \quad y = x + C e^{-x}$

Its family of orthogonal trajectories is given by the equation:
 * $x = y - 2 + C e^{-y}$


 * XplusCExpMinusXOrthogonalTrajectories.png

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

Differentiating $(1)$ $x$ gives:
 * $\dfrac {\d y} {\d x} = 1 - C e^{-x}$

Eliminating $C$:

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:
 * $-\dfrac {\d y} {\d x} = - x = 1 - y$

The integrating factor is $e^y$, giving:
 * $\ds e^y x = \int y e^y - e^y \rd y$

Using Primitive of $x e^{a x}$:
 * $\ds \int y e^y \rd y = y e^y - e^y$

Thus we get:
 * $e^y x = y e^y - e^y - e^y + C$

which gives us:
 * $x = y - 2 + C e^{-y}$