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Exponential decay is a fundamental concept often encountered in various scientific fields such as physics, biology, and finance. It describes a process where a quantity decreases at a rate proportional to its current value. In mathematical terms, it usually takes the form of an equation like:

• \( y = c e^{-kx} \)

where:

• \( y \) is the quantity that changes over time,
• \( c \) is a constant representing the initial amount,
• \( e \) is the base of the natural logarithm, and
• \( -k \) is the exponential rate of decay.

This results in a curve that rapidly decreases and then flattens as \( x \) increases. In the original exercise, this concept is represented by the equation \( y = c e^{-3x} \), suggesting a strong exponential decline. With every increase in \( x \), the value of \( y \) diminishes at an accelerating pace until it levels off.

Differential equations are equations that involve a function and its derivatives. They are essential tools for modeling real-world systems where change occurs over time. The process usually involves:

• Determining the rate of change of a variable.
• Understanding the relationship between different rates of change in a system.
• Using this understanding to predict future behavior.

In the case of orthogonal trajectories, we use the given family of curves' differential equation to identify a new set of curves that intersect the original ones at right angles. For example, the original exercise uses the differential equation

• \( \frac{dy}{dx} = -3y \)

From this, we derive the condition for orthogonality, \( m_1 \times m_2 = -1 \), leading to a new differential equation for orthogonal trajectories:

• \( \frac{dy}{dx} = \frac{1}{3y} \)

Solving this results in the trajectory curves. This process involves rearrangement and integration to find solutions like \( \frac{3}{2}y^2 = x + A \), giving us insight into the nature of the intersections.

Parabolic curves are an important family of conic sections characterized by a U-shaped symmetry. They can take different orientations depending on the equation. For a parabola defined as \( y^2 = 4ax \), the curve opens to the right if \( x \) is positive, and to the left if \( x \) is negative. These curves can be represented by the general equation of a parabola:

• \( ay^2 + bx + cy = d \)

In the context of the original exercise, the orthogonal trajectories are represented by the equation \( \frac{3}{2}y^2 = x + A \). Here:

• \( y^2 \) correlates with the parabolic form,
• \( A \) is a constant that determines the position of the parabola along the x-axis.

These parabolic curves serve as intersections with the original exponential decay curves. They illustrate a family of new paths -- providing a visual understanding of how these curves relate as orthogonal trajectories. As each family of curves affects the other, recognizing their role in orthogonal relationships aids in a deeper grasp of geometry and calculus.