Find orthogonal trajectories of the family of curves. identify orthogonal family y=ce^x

1.  Find dy/dx for the given family:  dy/dx = ce^x

2.  Take the negative reciprocal of this result:  (-1/c)e^x

3.  Integrate this last result with respect to x:    (-1/c)e^x

    This last result is the desired orth traj of the family of curves y = ce^x.

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: