Finding Oblique Trajectories to y=x-1+c*e^-x

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The discussion focuses on finding oblique trajectories to the family of curves defined by the equation y = x - 1 + c*e^-x. The first step involves deriving the differential equation by differentiating the equation with respect to x and eliminating the constant c. The slope of the curves at any point is represented by dy/dx. To determine the angle of intersection between two curves, the tangent angle formula tan(A+B) = (tanA + tanB) / (1 - tanAtanB) is utilized, requiring specific values for tanA and tanB.

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How do I find the oblique trajectories to the following family of curves:

y = x-1 + c*e^-x
 
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First find the differential equation to the family of curves given by differentiation wrt x and eliminating c. dy/dx will give slope of the curves at any point. If another curve's tangent makes an angle 'A', with this curve whose tangent at point of intersection is at an angle 'B', using tan formula:
tan(A+B)= (tanA+tanB)/(1-tanAtanB)

What are tanA and tanB to be substituted with next?
 

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