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Differential equations are mathematical equations that involve derivatives of a function. They describe relationships involving rates of change and are key tools in modeling real-world phenomena. Differential equations are prevalent in fields like physics, engineering, and economics. When you encounter a differential equation, you're dealing with a formula that expresses how a certain quantity changes with respect to another.

In the given exercise, the differential equation is \( x^{2} y'' - 7x y' + 15y = 0 \). This equation is called a second-order linear differential equation because it involves the second derivative of \( y \), designated as \( y'' \). Solving differential equations often involves finding a function or set of functions that satisfy the equation. The solution might vary depending on whether non-constant or constant solutions are considered.

A constant function is one of the simplest forms of a function in mathematics. It is denoted as \( y = c \), where \( c \) is a constant value. This means that no matter the input, the output remains the same, perfectly flat and unchanging over an interval.

• Constant functions have a constant rate of change, explicitly indicating that they do not change!
• Their graphical representation is a horizontal line on a Cartesian plane.
• An example of a constant function would be \( y = 3 \), which is simply represented as a straight line parallel to the x-axis.

Constant functions are integral in understanding solutions to equations as they embody solutions where no change occurs. As seen in the step-by-step solution, analyzing constant functions involves setting derivatives to zero to verify their characteristics.

In calculus, taking the derivative of a function helps to determine how that function changes. When a derivative is zero, \( y' = 0 \), it indicates no change. In other words, the function is stable at that point, operating as a constant function.

For a constant function, both the first derivative \( y' \) and the second derivative \( y'' \) are zero because the function's rate of change does not vary. This characteristic is essential when working with differential equations, as substituting zero derivatives into the equation can reveal whether constant solutions are possible understandings of the setup.
These zero derivatives represent the concept of equilibrium in many applications, signifying a state where forces balance and no net change occurs. This is why identifying and working with zero derivatives are crucial in confirming the existence of constant solutions for differential equations.