Problem 57 - Advanced Engineering Mathematics

Question

Consider the differential equation \(d y / d x=y(a-b y)\), where \(a\) and \(b\) are positive constants. (a) Either by inspection, or by the method suggested in Problems 33-36, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the \(y\)-axis on which a nonconstant solution \(y=\phi(x)\) is increasing. On which \(y=\phi(x)\) is decreasing. (c) Using only the differential equation, explain why \(y=a / 2 b\) is the \(y\)-coordinate of a point of inflection of the graph of a nonconstant solution \(y=\phi(x)\). (d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the \(x y\)-plane into three regions. In each region, sketch the graph of a nonconstant solution \(y=\phi(x)\) whose shape is suggested by the results in parts (b) and (c).

Step-by-Step Solution

Answer

Constant solutions: \(y=0\), \(y=\frac{a}{b}\). Increasing: \(0\frac{a}{b}\). Inflection point at \(y=\frac{a}{2b}\).

Step 1: Identify Constant Solutions

The given differential equation is \(\frac{d y}{d x} = y(a - b y)\). A constant solution occurs when \(\frac{d y}{d x} = 0\). Set \(y(a - b y) = 0\) and solve for \(y\):\[ y(a - b y) = 0 \] implies \[y = 0 \] or \[a - b y = 0\]. Solving \(a - b y = 0\) gives \(y = \frac{a}{b}\). Thus, the two constant solutions are \(y = 0\) and \(y = \frac{a}{b}\).

Step 2: Determine Intervals of Increase and Decrease

Examine the sign of \(\frac{d y}{d x} = y(a - b y)\) to determine where \(y = \phi(x)\) is increasing or decreasing.- The function is increasing when \(\frac{d y}{d x} > 0\), which occurs when \(a - b y > 0\) and \(y > 0\), giving \(0 < y < \frac{a}{b}\).- The function is decreasing when \(\frac{d y}{d x} < 0\), which occurs when \(a - b y < 0\) and \(y > \frac{a}{b}\).

Step 3: Identify Point of Inflection

For \(y = \phi(x)\), a point of inflection occurs where the concavity changes, or the second derivative is zero. Take the derivative of \(\frac{d y}{d x} = y(a - b y)\):\[\frac{d^2 y}{d x^2} = \frac{d}{dx}(y(a - b y)) = a\frac{d y}{d x} - 2b y\frac{d y}{d x}.\]When \(y = \frac{a}{2b}\), \(a - 2by = 0\), giving a zero second derivative, indicating a change in concavity.

Step 4: Sketch Constant Solutions and Nonconstant Solution Behavior

On a graph, draw horizontal lines for the constant solutions \(y = 0\) and \(y = \frac{a}{b}\). These lines divide the plane into three regions:- Below the line \(y = 0\): The solution \(y = 0\) is stable, and solutions approach this line as \(x\) increases.- Between \(y = 0\) and \(y = \frac{a}{b}\): Solutions start underneath \(y = \frac{a}{b}\), rise to a peak near \(y = \frac{a}{2b}\), and come back down towards \(y = 0\).- Above \(y = \frac{a}{b}\): Solutions decrease and asymptotically approach \(y = \frac{a}{b}\).

Key Concepts

Constant Solutions

In the realm of differential equations, a constant solution is a specific value of the dependent variable that satisfies the differential equation without changing over time. For the given equation \(\frac{d y}{d x} = y(a - b y)\), a constant solution is found by setting \(\frac{d y}{d x} = 0\). This situation implies no change in \(y\) with respect to \(x\).

• We see that \(y(a-b y) = 0\) is the equation to solve for constant solutions.
• The equation splits into \(y = 0\) or \(a - b y = 0\).
• Solving \(a - b y = 0\) results in \(y = \frac{a}{b}\).

Thus, the two constant solutions are \(y = 0\) and \(y = \frac{a}{b}\), which help us understand the equilibrium states of this system.

Intervals of Increase and Decrease

Determining intervals of increase or decrease is crucial for understanding how a function behaves over different regions on its graph. For the nonconstant solution \(y = \phi(x)\) of the differential equation \(\frac{dy}{dx} = y(a - b y)\), we consider where the function's derivative is positive or negative.

• To identify increasing intervals, we examine where \(\frac{dy}{dx} > 0\). This condition holds when \(y(a - b y) > 0\).
• It implies the inequality \(a - b y > 0\) and \(y > 0\), leading to the interval \(0 < y < \frac{a}{b}\) for increasing behavior.
• Conversely, \(\frac{dy}{dx} < 0\) indicates decreasing intervals, which occur where \(a - b y < 0\) with \(y > \frac{a}{b}\).

These conclusions allow us to anticipate how the graph of \(y = \phi(x)\) changes as \(y\) increases or decreases.

Point of Inflection

A point of inflection is a location on a graph where the concavity changes, signifying a transition from a concave up to a concave down shape or vice versa. To identify a point of inflection for \(y = \phi(x)\), we explore where the second derivative changes sign.

• We find this by taking the derivative of \(\frac{dy}{dx} = y(a - b y)\).
• Differentiating gives \(\frac{d^2y}{dx^2} = a\frac{dy}{dx} - 2b y\frac{dy}{dx}\).
• Setting \(\frac{d^2y}{dx^2} = 0\) yields that an inflection point occurs where \(y = \frac{a}{2b}\).

At this point, \(a - 2by = 0\) holds, indicating a change in concavity. Understanding this concept is key to sketching accurate graphs of the solutions.

Solution Behavior Analysis

Analyzing the behavior of solutions provides insight into how solutions to the differential equation behave over different regions on a graph. This involves sketching graphs and predicting the trends of the solutions based on constant solutions and intervals of increase or decrease.

• The constant solutions \(y = 0\) and \(y = \frac{a}{b}\) split the graph into distinct regions.
• Below the line \(y = 0\), the graph of solutions is attracted toward this stable line.
• In the region between \(0 < y < \frac{a}{b}\), solutions rise, peaking near \(y = \frac{a}{2b}\), then slope back towards \(y = 0\).
• Above \(y = \frac{a}{b}\), solutions tend to decrease and approach \(y = \frac{a}{b}\) asymptotically.

This comprehensive analysis tells us how solutions migrate or stabilize in different intervals, aiding in sketching a complete picture of solution behavior.