Please help me out with differential equations

Question

Find the orthogonal trajectories of the following family of curves

y=cx/1+x

please write the solution with full steps

Answer 1

To find the orthogonal trajectories, you'll have to take the first derivative of the function, take the inverse recipricol of dy/dx, then solve for that differential equation.

1. Given y = (cx)/(1+x), and taking the derivative of the function yields, dy/dx = c/(1+x)**2,
2. The inverse recipricol is therefore, dy/dx(ortho) = -(1+x)**2/c,
3. Solving the above differential equation reveals the solution, -((x+1)**3)/3c
4. Solution: Orthoganal Trajectory Family of Curves O(x) = -((x+1)**3)/3c <<<-------- Solution