Problem 57 - Advanced Engineering Mathematics

Question

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems \(55-58\), recall the geometric significance of the derivatives \(d y / d x\) and \(d^{2} y / d x^{2}\). 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 thedifferentialequation, 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 are \( y=0 \) and \( y=\frac{a}{b} \). Increasing when \( 0\frac{a}{b} \). Inflection at \( y=\frac{a}{2b} \).

Step 1: Find Constant Solutions

To find constant solutions, we look for values of \( y \) such that the derivative \( \frac{dy}{dx} = 0 \). Given the differential equation \( \frac{dy}{dx} = y(a - by) \), set the right-hand side equal to zero: \( y(a - by) = 0 \). Solving \( y(a - by) = 0 \) gives two solutions: \( y = 0 \) and \( y = \frac{a}{b} \). Thus, the constant solutions are \( y = 0 \) and \( y = \frac{a}{b} \).

Step 2: Determine Intervals of Increase or Decrease

To determine when \( y = \phi(x) \) is increasing or decreasing, we analyze \( \frac{dy}{dx} = y(a - by) \). The solution is increasing when \( \frac{dy}{dx} > 0 \) and decreasing when \( \frac{dy}{dx} < 0 \). Calculate: \( y(a - by) > 0 \) implies \( 0 < y < \frac{a}{b} \), meaning \( y = \phi(x) \) is increasing when \( y \) is in this interval. For \( y(a - by) < 0 \), it implies \( y > \frac{a}{b} \), so \( y = \phi(x) \) is decreasing when \( y > \frac{a}{b} \).

Step 3: Identify Points of Inflection

To find the point of inflection, consider the second derivative \( \frac{d^2 y}{dx^2} \). Differentiate \( \frac{dy}{dx} = y(a - by) \) with respect to \( x \): \( \frac{d^2y}{dx^2} = \frac{d}{dx}[y(a-by)] = \frac{dy}{dx}(a-by) + y(-b\frac{dy}{dx}) \). At \( y = \frac{a}{2b} \), the term \( a - 2by = 0 \), which causes the concavity to change, indicating a point of inflection. Therefore, \( y = \frac{a}{2b} \) is the \( y \)-coordinate of a point of inflection.

Step 4: Sketch Graphs

On a coordinate plane, plot the constant solutions \( y = 0 \) and \( y = \frac{a}{b} \). These lines divide the plane into three regions. For \( 0 < y < \frac{a}{b} \), draw an increasing curve. For \( y > \frac{a}{b} \), draw a decreasing curve. The point \( y = \frac{a}{2b} \) marks the location of an inflection point, showing a change in concavity. Sketch the nonconstant solutions accordingly, reflecting their behavior in these regions.

Key Concepts

Constant Solutions

In differential equations, constant solutions are solutions where the function does not change with respect to the independent variable, which means its derivative is zero. In our given differential equation \( \frac{dy}{dx} = y(a - by) \), setting the derivative to zero \( y(a - by) = 0 \) allows us to solve for constant values of \( y \). This leads to either \( y = 0 \) or \( y = \frac{a}{b} \) as solutions.
These solutions are particularly important because they act as equilibrium points where the rate of change is zero, and thus the system remains at these states if initiated from them. They represent constants that do not vary with different values of \( x \), meaning a graph of these solutions would be horizontal lines at \( y = 0 \) and \( y = \frac{a}{b} \) in the coordinate plane.

Intervals of Increase/Decrease

Finding the intervals where a solution is increasing or decreasing involves analyzing the sign of the derivative. Our equation \( \frac{dy}{dx} = y(a - by) \) shows that the solution \( y = \phi(x) \) is increasing where \( \frac{dy}{dx} > 0 \). This happens when \( y(a - by) > 0 \).
By solving \( 0 < y < \frac{a}{b} \), we discover that the solution is increasing when \( y \) is within this range. Conversely, the solution decreases when \( \frac{dy}{dx} < 0 \), or \( y > \frac{a}{b} \).

• For \( 0 < y < \frac{a}{b} \), the curve of the solution is rising.
• For \( y > \frac{a}{b} \), the curve descends.

Understanding these intervals helps in sketching the trajectory of a nonconstant solution across the defined spaces in the graph.

Points of Inflection

Points of inflection are important features of a graph where the curvature changes, which is usually identified by the second derivative. In our problem, the second derivative is key to finding points of inflection along the solutions of the differential equation \( \frac{dy}{dx} = y(a - by) \).
By calculating the second derivative \( \frac{d^2 y}{dx^2} \) and analyzing it for changes in sign, we find that at \( y = \frac{a}{2b} \), the term \( a - 2by = 0 \) holds true, indicating a change in concavity.

• This inflection point highlights a shift from concave up to concave down or vice versa in our solution graph.

Recognizing the inflection point is crucial as it dictates the shape of the nonconstant solutions as they pass through this significant coordinate.

Solution Sketching

Sketching solutions to a differential equation allows visualization of how solutions traverse through the mathematical landscape described by the equation. For our scenario:\( \frac{dy}{dx} = y(a - by) \), sketching involves plotting constant solutions and the behaviors of nonconstant solutions.

• First, draw horizontal lines at \( y = 0 \) and \( y = \frac{a}{b} \) to represent the constant solutions.
• The space between these lines, \( 0 < y < \frac{a}{b} \), shows where solutions are increasing, so sketch curves that move upwards in this region.
• Above \( y = \frac{a}{b} \), solutions are decreasing, so draw curves that slope downwards beyond this line.

The point \( y = \frac{a}{2b} \) serves as a landmark for an inflection point, indicating a shift in curvature, providing a comprehensive picture of how solutions behave in different regions of the graph.