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

Eliminating $C$:

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

The integrating factor is $e^y$, simply enough, giving:
 * $\displaystyle e^y x = \int y e^y - e^y \, \mathrm d y$

Using Primitive of $x e^{a x}$:
 * $\displaystyle \int y e^y \, \mathrm d y = y e^y - e^y$

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

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