Explanation:
The given curve is represented by the equation y^2 = 3x^3 + cx, where c is a constant.

Finding the Orthogonal Trajectories:
To find the orthogonal trajectories of the given curve, we need to determine the differential equation satisfied by these trajectories.

Step 1:
Differentiate the given equation with respect to x to find the slope of the tangent line to the curve.
2yy' = 9x^2 + c

Step 2:
Determine the slope of the line perpendicular to the tangent line.
The slope of the perpendicular line is given by -1/m, where m is the slope of the tangent line.

Step 3:
Replace y' with -1/m in the equation obtained in Step 1.
2y(-1/m) = 9x^2 + c
Simplifying the equation, we get:
y = (-m/2)(9x^2 + c)

Step 4:
Differentiate the equation obtained in Step 3 with respect to x to find the differential equation satisfied by the orthogonal trajectories.
dy/dx = (-m/2)(18x)

Step 5:
Replace m with -1/m in the equation obtained in Step 4.
dy/dx = -1/2m(18x)

Step 6:
Simplify the equation further to obtain the differential equation of the orthogonal trajectories.
dy/dx = -9x/m

Step 7:
To eliminate the parameter m, we express it in terms of x and y by using the equation y^2 = 3x^3 + cx.
y^2 = 3x^3 + cx
Differentiating both sides with respect to x, we get:
2yy' = 9x^2 + c
Simplifying the equation, we obtain:
y' = (9x^2 + c)/(2y)

Step 8:
Replace y' with -1/m in the equation obtained in Step 7.
-1/m = (9x^2 + c)/(2y)

Step 9:
Simplifying the equation, we get:
2y/m = -(9x^2 + c)

Step 10:
Multiply both sides of the equation by m, we obtain:
2y = -m(9x^2 + c)

Step 11:
Rearrange the equation to express m in terms of x and y.
m = -2y/(9x^2 + c)

Step 12:
Substitute the expression for m obtained in Step 11 into the equation obtained in Step 6.
dy/dx = -9x/(-2y/(9x^2 + c))
Simplifying the equation, we get:
dy/dx = 9x(9x^2 + c)/(2y)

Step 13:
Simplifying further, we obtain:
dy/dx = (81x^3 + 9cx)/(2y)

Step 14:
Multiply both sides of the equation by 2y to eliminate