Question

Increasing Function In Exercises 27 and 28 , determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x y $$

Short answer

The solution of the differential equation is an increasing function in the first and third quadrants.

Step-by-step solution

Consider the given derivative

The given derivative is \(\frac{dy}{dx} = \frac{1}{2}xy\). This result corresponds to the rate of change of \(y\) w.r.t \(x\).

Determine the condition for an increasing function

We know that a function is increasing when its derivative is positive. Therefore, we need \(\frac{1}{2}xy > 0\) for the function to be increasing.

Find the quadrants where the function is increasing.

Remember that in the Cartesian system, in the first quadrant both \(x\) and \(y\) are positive, making the product of \(x\) and \(y\) positive. Similarly, in the third quadrant, both \(x\) and \(y\) are negative, which also results in a positive product. Hence, \(y(x)\) should be an increasing function in the first and third quadrants, as \(\frac{dy}{dx} > 0\) in these quadrants.

Key concepts

Differential Equations

Differential equations play a fundamental role in calculus and various applied fields, as they describe the relationship between a function and its derivatives. From predicting how populations change to modeling the motion of heavenly bodies, differential equations offer a way to express how quantities vary in relation to each other.

For instance, in our exercise, the differential equation \( \frac{dy}{dx} = \frac{1}{2}xy \) encapsulates the rate at which the dependent variable \( y \) changes with respect to the independent variable \( x \). The equation suggests that the rate of change isn't constant but varies depending on the current values of \( x \) and \( y \). This is typical of many natural processes described by differential equations.

When understanding differential equations, it's crucial to consider the signs and values the variables \( x \) and \( y \) can take, as they determine the behavior of the solution — for example, dictating when a function is increasing or decreasing. Solving these equations often involves integrating factors, separation of variables, or using numerical methods when an analytical solution isn't possible.

First and Third Quadrants

The Cartesian coordinate system is divided into four quadrants. The first quadrant is where both \( x \) and \( y \) are positive, while in the third quadrant, both \( x \) and \( y \) are negative. This distinction is vital for determining the behavior of functions defined by differential equations.

When the differential equation from our exercise, \( \frac{dy}{dx} = \frac{1}{2}xy \), is considered, the sign of the product \( xy \) dictates the function's growth. Since the product of two numbers with the same sign is always positive, the rate of change \( \frac{dy}{dx} \) will be positive in both the first and third quadrants. This means that if the function \( y(x) \) follows this differential equation, it will be increasing in both quadrants. Understanding which quadrant you're dealing with is crucial for interpreting the behavior of solutions to differential equations, as it sets the stage for whether the function is increasing or decreasing—or in other contexts, whether a system is gaining or losing energy.

Rate of Change

The concept of 'rate of change' is fundamental to understanding how variables evolve over time or in relation to each other. In calculus, it's often represented as a derivative. The derivative \( \frac{dy}{dx} \) tells us how fast \( y \) is changing with respect to \( x \). If the derivative is positive, \( y \) is increasing as \( x \) increases; if negative, \( y \) is decreasing.

In the context of the exercise, the given differential equation \( \frac{dy}{dx} = \frac{1}{2}xy \) gives us the rate of change of \( y \) with respect to \( x \) directly tied to the values of \( x \) and \( y \) themselves. When analyzing this rate of change without solving the equation, we infer that for \( y(x) \) to be increasing—meaning that \( \frac{dy}{dx} > 0 \)—the product \( xy \) must be positive. This relationship provides a stepping stone in understanding more complex scenarios where variables interact in intricate ways that are modeled through differential equations.