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.) $$ \frac{d y}{d x}=\frac{1}{2} x^{2} y $$

Short answer

The solution of the given differential equation is an increasing function in Quadrants I and II.

Step-by-step solution

Understand the Given Differential Equation

The differential equation given is \(\frac{d y}{d x}=\frac{1}{2} x^{2} y\). The derivative of \(y\) with respect to \(x\) is equal to one-half of the product of the square of \(x\) and \(y\). Since the product of \(x^{2}\) and \(y\) influences the rate of change of \(y\), the solution of the equation will be an increasing function where this product is positive.

Determine the Sign of the Derivative

The derivative \(\frac{d y}{d x}\) will be positive where \(\frac{1}{2} x^{2} y\) is positive. Since \(x^{2}\) is always nonnegative for real \(x\), the sign of \(\frac{1}{2} x^{2} y\) depends on the sign of \(y\). When \(y\) is positive, the derivative is positive. Therefore, the solution is an increasing function in the regions where \(y\) is positive.

Determine the Quadrants

Looking at the four quadrants: Quadrant I (\(x>0, y>0\)), Quadrant II (\(x<0, y>0\)), Quadrant III (\(x<0, y<0\)), and Quadrant IV (\(x>0, y<0\)), the solution \(y\) is positive in Quadrants I and II. Hence, the solution is an increasing function in Quadrants I and II.

Key concepts

Differential Equations

Differential equations are mathematical expressions that describe the relationship between a function and its derivatives. In simple terms, they show us how a particular quantity changes over time or space, which is of great importance in various fields such as physics, engineering, economics, and biology. In the context of the given exercise, we have the differential equation \( \frac{d y}{d x}=\frac{1}{2} x^{2} y \). This particular kind of differential equation is known as a first-order equation because it involves only the first derivative of the function \(y\).

The equation relates the rate of change of the function \(y\) with respect to \(x\) to \(y\) itself and the square of \(x\). Understanding differential equations allows us to predict the behavior of dynamical systems, model natural phenomena, and solve practical problems in various scientific domains.

Function Behavior Analysis

The behavior of a function is fundamentally about understanding how the function increases or decreases over its domain. Function behavior analysis focuses on evaluating the function's growth or decay, which can be inferred from the signs of its derivatives. In the given exercise, by analyzing the sign of the differential equation \( \frac{d y}{d x}=\frac{1}{2} x^{2} y \), we deduced where the function \(y\) is increasing.

Since the derivative represents the rate of change, a positive derivative indicates an increasing function. When \(y\) and \(x^{2}\) are both positive, their product, and consequently the derivative, is positive, signaling that the function is increasing. This straightforward analysis is a powerful tool in understanding the essential nature of the function without solving the differential equation explicitly.

Quadrant Determination

Quadrant determination is a method used to analyze the signs of values within the Cartesian coordinate system. The coordinate system is divided into four quadrants based on the signs of the \(x\) and \(y\) axes. In Quadrant I, both \(x\) and \(y\) are positive. In Quadrant II, \(x\) is negative and \(y\) is positive. Quadrant III has both \(x\) and \(y\) as negative, and in Quadrant IV, \(x\) is positive and \(y\) is negative.

In our exercise, we're interested in the quadrants where the function \(y\) is an increasing function, which occurs when its derivative is positive. From our analysis, it's clear that this condition is met in Quadrants I (where positive \(x\) and \(y\) ensure a positive product \(x^{2}y\)) and II (where although \(x\) is negative, the square \(x^{2}\) ensures a nonnegative factor, and a positive \(y\) results in a positive derivative). Therefore, the solution is an increasing function in these quadrants, illustrating the importance of quadrant determination in analyzing the behavior of functions without the need for complete solutions to the differential equations.