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A differential equation is an equation that relates a function to its derivatives. These equations are fundamental in understanding how dynamic systems change over time. In our specific problem, we are dealing with the differential equation that comes from the family of circles which are tangent to the y-axis at the origin.
This involves finding the derivative of a function that describes these circles, allowing us to derive the relationship between their orthogonal trajectories.
By working with a differential equation, we can find solutions that describe how a quantity changes with respect to some variable like time or space. In our case, we are looking at spatial variables, so the differential equation helps us find curves (trajectories) that cut the original curves at right angles.

Implicit differentiation is a technique used when it is difficult or impossible to solve equations for one variable in terms of another before differentiating.
This type of differentiation is particularly useful when we handle equations involving circles or other non-linear objects.
For the circles tangent to the y-axis, the equation is \( (x-r)^2 + y^2 = r^2 \). Differentiating such equations implicitly involves treating all terms involving a variable like y as functions of x, allowing us to find the rate at which y changes with x.
Implicit differentiation becomes crucial in this exercise to find the slope of the curves that are orthogonal to the original family of circles.

When we talk about circles tangent to the y-axis at the origin, it means these circles just "touch" the y-axis without crossing it at the origin point.
The standard circle equation is \( (x-h)^2 + (y-k)^2 = r^2 \), indicating a circle with center at \((h, k)\) and radius \(r\).
For a circle tangent to the y-axis at the origin, the center must lie on the x-axis. This is because the y-axis acts as a line of symmetry, and touching means at zero distance horizontally. Therefore, the center is \( (r, 0) \), with "r" being the radius - the distance from the circle's center to its tangent point.

The concept of an orthogonal derivative involves determining the slopes of curves that are perpendicular to a given family of curves at every point of intersection.
This is foundational for finding orthogonal trajectories, which are curve sets that intersect at right angles with each member of the original family of curves.
Simplifying, for any function, if the derivative \( \frac{dy}{dx} \) describes the slope, its orthogonal trajectory will have slope \(-\frac{dx}{dy}\). This transformation gives us a new differential equation representing the orthogonal trajectories.
By using implicit differentiation and eliminating parameters as shown in the original steps, we derived the orthogonal slopes leading towards determining these unique intersecting paths.