Removing parameter from set of orthogonal trajectories

So during the process of finding a solution I considered the first integral constant to be zero to try and simplify the solution. It turns out The family of solutions is not complete in turns of their ability to describe all solutions to the differential equation.

I have solved the equation by considering that integral constant below and was able to remove the parameter by substitution. Thanks to everyone who tried to help!! Hope this helps anyone who finds this thread.

$$frac{dy}{dx} = sqrt{frac{c}{x}}$$

integrating with respect to x: $$ y= 2sqrt{cx} + A$$ where A is the first integral constant. Let this be Equation 1

Without considering A=0 like I did, Rearrange the above expression to get a suitable substitution:

$$ sqrt{c} = frac{y-A}{2sqrt{x}} $$

Substituting in equation 1 we get:

$$ frac{dy}{dx} = frac{y-A}{2x} $$

Therefore the differential equation for the orthogonal trajectories:

$$ frac{dy}{dx} = frac{-2x}{y-A} $$

solving this we get the final solution:

$$ y^2 + 2x^2 -2Ay = B $$

where B is the second integral constant.

Here's a desmos graph to play around with. Enjoy! https://www.desmos.com/calculator/ppwwgeglos