Problem 38 - Advanced Engineering Mathematics

Question

For purposes of this problem, ignore the graphs given in Figure 5.1.1. If Airy's \(\mathrm{DE}\) is written as \(y^{\prime \prime}=-x y\), what can we say about the shape of a solution curve if \(x>0\) and \(y>0 ?\) If \(x>0\) and \(y<0 ?\)

Step-by-Step Solution

Answer

If \(x > 0\) and \(y > 0\), the solution curve is concave down; if \(x > 0\) and \(y < 0\), it is concave up.

Step 1: Understand the Differential Equation

The given differential equation is Airy's DE, written as \( y'' = -x y \). Here, \( y'' \) represents the second derivative of \( y \) with respect to \( x \), and \( -x y \) represents the product of \( x \) and \( y \) multiplied by -1. The problem asks us to analyze the shape of the solution curve under certain conditions.

Step 2: Analyze the effect of \(x > 0\) and \(y > 0\)

In this scenario, \( x > 0 \) and \( y > 0 \), making \( -x y < 0 \). This implies that \( y'' < 0 \). A negative second derivative indicates that the curve is concave down. Therefore, the solution curve is bending downwards as \( x \) increases.

Step 3: Analyze the effect of \(x > 0\) and \(y < 0\)

Here, we have \( x > 0 \) and \( y < 0 \), leading to \( -x y > 0 \). This means that \( y'' > 0 \). A positive second derivative indicates that the curve is concave up. Thus, the solution curve is bending upwards as \( x \) increases.

Key Concepts

Second Derivative

In the study of differential equations, the second derivative, denoted as \( y'' \), plays a crucial role in understanding the behavior of a solution curve. The second derivative is simply the derivative of the first derivative and provides us with information about the rate of change of the slope of a function at any point.
When dealing with Airy's Differential Equation, \( y'' = -x y \), the second derivative reveals how the curve is bending.

• If \( y'' > 0 \), the curve is bending upwards.
• If \( y'' < 0 \), the curve is bending downwards.

This information is vital for predicting the shape of the solution curve in response to various values of \( x \) and \( y \) and understanding how the function behaves over its domain.

Concavity

Concavity relates directly to the second derivative and gives us insights into the shape and bending of the curve. When analyzing differential equations, such as the given Airy's Differential Equation, identifying concavity is imperative.
- If the second derivative \( y'' < 0 \), the curve is concave down. This means the curve looks like an upside-down bowl or arch, bending downwards as seen with \( x > 0 \) and \( y > 0 \).
- Conversely, if \( y'' > 0 \), the curve is concave up, indicating that it resembles a "U" shape, bending upwards as in the case where \( x > 0 \) and \( y < 0 \).
Understanding concavity is crucial in sketching and interpreting how the solution curve behaves, responding to different conditions given in the problem.

Solution Curve

The solution curve is the graphical representation of the solutions to a differential equation. In Airy's Differential Equation system, where we've defined \( y'' = -x y \), the solution curve changes based on the conditions of \( x \) and \( y \).
For \( x > 0 \):

• If \( y > 0 \), the second derivative is negative, and thus the curve bends downwards, indicating a concave-down nature.
• If \( y < 0 \), the second derivative is positive, making the curve bend upwards and exhibit a concave-up shape.

These solution curves visually depict how the relationship of \( x \) and \( y \) influences the direction and bending of the graph, helping us predict behavior over time.

Differential Equations Analysis

Analyzing differential equations involves understanding relationships between variables and their derivatives. Airy's Differential Equation given by \( y'' = -x y \), introduces an opportunity to explore how solutions evolve.
Such equations require one to:

• Identify the second derivative to assess the curve's shape.
• Determine how changes in \( x \) and \( y \) influence the concavity and form of the solution curve.
• Predict the graphical representation based on mathematical properties.

By thoroughly dissecting each component such as what happens to \( y'' \) when \( x > 0 \) and either \( y > 0 \) or \( y < 0 \), students can gain a deeper understanding of the behavior of differential equations. This approach underpins much of the practical analysis in fields like physics and engineering where solving differential equations is essential.