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In mathematics, differential equations play a crucial role in describing various phenomena, from physics to engineering. A differential equation involves derivatives, which represent dynamic changes in systems.
For example, if you are analyzing how an object moves, a differential equation can help describe its velocity and acceleration over time. When we deal with families of curves like those given in the exercise, differential equations help us find orthogonal trajectories. Orthogonal trajectories are curves that intersect the given family of curves at right angles. Think about it as drawing a new path that stylishly weaves through the existing paths at 90-degree angles.
To find these orthogonal trajectories, you start by finding the derivative of the original family of curves. This derivative represents the slope of the originally given curves. For a new curve to intersect at right angles, its derivative must be the negative reciprocal of this slope.
In simpler terms, if the slope of the original curve is represented as A/B, the orthogonal curve will have a slope of -B/A. In the solution, this step is crucial to deriving the differential equation for orthogonal trajectories from the original curve equations given in the exercise.

Curve sketching is like giving life to mathematical equations by drawing their visuals on paper. It allows us to see the behavior and properties of different functions and solve problems more intuitively.
When sketching curves from families like the ones discussed in the exercise, we first identify their general shapes. For instance, the equation \(xy = c\) describes a family of hyperbolas. Hyperbolas have two branches and open in opposite directions. Identifying such properties is essential before drawing. Once you have plotted the general shape of the original family, you need to add the orthogonal trajectories. As shown in the solution, these end up being another set of hyperbolas for our case. What’s exciting here is how these intersect the original curves perpendicularly, maintaining geometrical harmony. Curve sketching becomes a powerful tool to visualize these intersections and understand their significance.
When you sketch both families of curves, you create a delicate mesh that beautifully combines both shape and orthogonal intersections. This visual step helps cement the mathematical principles in a more concrete form.

Integration is a fundamental tool in calculus, used for various purposes like finding area under curves. In the context of orthogonal trajectories, it is used to solve differential equations derived from the original curve equations.
After determining the differential equation for the orthogonal trajectories, you'll often reach a step where you'll have to integrate both sides to solve it. For instance, in our example, the equation \(y\,dy = x\,dx\) is separated so that we can integrate the left side with respect to \(y\) and the right side with respect to \(x\). This step transforms the differential equation back into a form involving just the original variables. The integration results, \(\frac{y^2}{2} = \frac{x^2}{2} + C\), give us the equation of the orthogonal trajectories. This tells us the form of curves that meet the original family at right angles.
In simple terms, the integration process is like putting together puzzle pieces—combining small changes (the derivatives) into a complete picture (the equation of the trajectory). Using these integration techniques ensures that we can correctly and accurately map out the relationships between different mathematical quantities.