Calculus - Orthogonal Trajectories

Find the orthogonal trajectories of the family of curves:

y = k*(e^-x)
so k = y/(e^-x)

differentiating we get:

1 = -k(e^-x)*(dx/dy)
1/(dx/dy) = -k(e^-x)
dy/dx = -k(e^-x)...substituting for k:
dy/dx = -(y/(e^-x))*(e^-x)
dy/dx = -y
Integral(1/=y)dy = Integral dx
-ln(y) = x + C
x = -ln(y) + C

Is this correct? Thanks.

Yes, correct.

Yes, correct.

Thanks for checking!!!

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