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Ordinary differential equations (ODEs) are mathematical equations that relate functions and their derivatives. They are used to describe the changing rates of various phenomena, from simple motion to the dynamics of populations. A single ODE or a system of ODEs can model a wide range of physical, biological, economic, and engineering processes.

When solving an ODE, like the one in our exercise, we apply specific techniques such as separation of variables or integrating factors. In our textbook example, we examined the family of parabolas defined by the equation \(y^{2}=2 c x+c^{2}\). To understand the behaviour of these curves, we first derived their differential equation by differentiating with respect to \(x\), which provided a relationship between \(y\), \(dy/dx\), and the parameter \(c\).

After obtaining the differential equation, we performed algebraic manipulations to express it in a standard form suitable for finding solutions or analyzing further properties. This step is fundamental in understanding the broader concept of ODEs and how they govern the systems they represent.

A family of parabolas is a set of curves that can be described by varying a parameter within a general equation. In the context of our exercise, the family of parabolas is given by \(y^{2}=2 c x+c^{2}\), where \(c\) is a parameter that can take on any real value. Each value of \(c\) defines a different member of the parabola family.

These parabolas share a common set of features, which include a symmetrical shape, a vertex that shifts based on the value of \(c\), and an axis of symmetry. In understanding the family of parabolas, we can analyze how the curves move and evolve as the parameter changes.

When facing such a family of curves, a Differential Equations approach lets us create a unifying description of the entire set through one encompassing equation. This method not only streamlines our understanding of the curves but also paves the way to discovering additional properties, such as self-orthogonality, which we explore in our main exercise.

Orthogonal trajectories are curves that intersect a given family of curves at right angles (90 degrees). These trajectories have great significance in fields ranging from physics to geometry because they often represent contours with opposing physical properties or complementary relationships.

In our step-by-step solution, we sought the trajectories orthogonal to the given family of parabolas. To find these trajectories, we took the derivative of the original set of curves and applied a key idea: two lines are orthogonal if the product of their slopes is -1. This led us to replace the derivative \(dy/dx\) with \(-dx/dy\) in the differential equation from Step 1.

After solving the new differential equation, we obtained another set of curves. The determination that these new curves formed the same family of parabolas confirmed the self-orthogonality of the original family. Understanding orthogonal trajectories is essential because it allows us to explore how different system states or conditions can be intrinsically connected in a perpendicular and thus equilibrium-establishing manner.