Find an equation of orthogonal trajectories of the curve of the family 
y
=
x
−
1
+
c
e
−
x
?

To find the equation of the orthogonal trajectories of the given family of curves:

\[
y = x - 1 + c e^{-x}
\]

we follow these steps:

### Step 1: Differentiate the given equation with respect to \(x\)
First, differentiate the given equation with respect to \(x\):

\[
\frac{dy}{dx} = 1 - c e^{-x}
\]

This gives the slope of the tangent lines to the given family of curves.

### Step 2: Slope of the orthogonal trajectories
The slopes of the orthogonal trajectories will be the negative reciprocal of the slope of the given family of curves. So, the slope of the orthogonal trajectories is:

\[
\frac{dy}{dx}_{\text{orthogonal}} = - \frac{1}{1 - c e^{-x}}
\]

### Step 3: Eliminate the parameter \(c\)
From the original equation \( y = x - 1 + c e^{-x} \), solve for \(c\):

\[
c = e^{x}(y - x + 1)
\]

Substitute this into the expression for the orthogonal slope:

\[
\frac{dy}{dx}_{\text{orthogonal}} = - \frac{1}{1 - (y - x + 1)}
\]

Simplifying:

\[
\frac{dy}{dx}_{\text{orthogonal}} = - \frac{1}{2 - y + x}
\]

### Step 4: Solve the differential equation
Now solve the differential equation:

\[
\frac{dy}{dx} = - \frac{1}{2 - y + x}
\]

This is a separable differential equation. Rearranging terms:

\[
(2 - y + x) \, dy = -dx
\]

Integrate both sides:

\[
\int (2 - y + x) \, dy = - \int dx
\]

This leads to:

\[
\frac{(2 - y + x)^2}{2} = -x + C
\]

This equation describes the orthogonal trajectories of the given family of curves.

Would you like more details or clarifications? Here are 5 related questions:

1. What are orthogonal trajectories, and how are they applied in different fields?
2. How do we verify if two curves are orthogonal to each other at a given point?
3. Could you explain separable differential equations in more depth?
4. How would you find orthogonal trajectories for other types of curves?
5. Can this method of orthogonal trajectories be applied in 3D space?

**Tip:** When solving for orthogonal trajectories, always make sure to properly differentiate and eliminate constants to get a correct differential equation.

Find an equation of orthogonal trajectories of the curve of the family y = x - 1 + c e^{-x}?

Explore how to find the equation of orthogonal trajectories for the curve family y = x - 1 + c e^{-x}. This problem uses concepts of differential equations and separable equations, providing a step-by-step solution. Suitable for advanced high school or undergraduate math students.

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Find an equation of orthogonal trajectories of the curve of the family y

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