Problem 56 - 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 \(d y / d x=5-y\). (a) Either by inspection or by the method suggested in Problems \(33-36\), find a constant solution of the \(\mathrm{DE}\). (b) Using only the differential equation, find intervals on the \(y\) -axis on which a nonconstant solution \(y=\phi(x)\) is increasing. Find intervals on the \(y\) -axis on which \(y=\phi(x)\) is decreasing. (c) Explain why \(y=0\) is the \(y\) -coordinate of a point of inflection of a solution curve. (d) Sketch the graph of a solution \(y=\phi(x)\) of the differential equation whose shape is suggested by parts (a)-(c).

Step-by-Step Solution

Answer

(a) Constant solution: \(y = 5\). (b) Increasing for \(y < 5\), decreasing for \(y > 5\). (c) Not a point of inflection, but \(y = 5\) is an asymptotic solution.

Step 1: Find a Constant Solution

To find a constant solution to the differential equation \(\frac{dy}{dx} = 5 - y\), assume \(y = c\) where \(c\) is a constant. Then \(\frac{dy}{dx} = 0\). Substitute \(y = c\) into the equation: \[0 = 5 - c.\] Solve for \(c\) to find \(c = 5\). Therefore, the constant solution is \(y = 5\).

Step 2: Determine Increasing and Decreasing Intervals

Using the differential equation \(\frac{dy}{dx} = 5 - y\), we determine where the function is increasing or decreasing:- **Increasing**: \(\frac{dy}{dx} > 0\). Thus, \(5 - y > 0\) which implies \(y < 5\).- **Decreasing**: \(\frac{dy}{dx} < 0\). Thus, \(5 - y < 0\) which implies \(y > 5\).Therefore, \(y = \phi(x)\) is increasing on intervals where \(y < 5\) and decreasing on intervals where \(y > 5\).

Step 3: Identify the Point of Inflection

For inflection points, consider the second derivative. Starting from \(\frac{dy}{dx} = 5 - y\), differentiate both sides with respect to \(x\): \[\frac{d^2y}{dx^2} = -\frac{dy}{dx}.\]At the point of inflection, \(\frac{d^2y}{dx^2} = 0\). So \(-\frac{dy}{dx} = 0\), which implies \(\frac{dy}{dx} = 0\). Since \(\frac{dy}{dx} = 5 - y\), setting \(5 - y = 0\) gives \(y = 5\). Therefore, at \(y = 5\), the curve is not a point of inflection but rather where the concavity changes.

Step 4: Sketch the Solution Curve

Based on the analysis:- The constant solution \(y = 5\) is a horizontal line.- For \(y < 5\), \(\phi(x)\) is increasing and approaches \(y = 5\) as \(x\) increases.- For \(y > 5\), \(\phi(x)\) is decreasing, also approaching \(y = 5\).- Sketch a curve that starts below \(y = 5\), increases towards it, then decreases if starting above, approaching \(y = 5\) asymptotically as \(x\) goes to infinity.

Key Concepts

Constant Solutions

When solving differential equations, a constant solution is one where the derivative is zero over all its domain. This means that the solution does not change as the independent variable (often denoted as \( x \)) changes. For the differential equation \( \frac{dy}{dx} = 5 - y \), a constant solution can be found by setting \( y = c \), where \( c \) is a constant. Substituting this into the equation, we get \( 0 = 5 - c \). Solving this gives \( c = 5 \). Thus, the constant solution is \( y = 5 \).
In practice, this constant represents a horizontal line on the graph, and it's evident that no matter the value of \( x \), \( y \) remains 5. This is because the rate of change of \( y \) with respect to \( x \) is zero, signifying no growth or decline.

Increasing and Decreasing Intervals

Identifying where a function is increasing or decreasing is crucial in understanding its behavior. For the differential equation \( \frac{dy}{dx} = 5 - y \), the sign of \( \frac{dy}{dx} \) helps us understand these intervals.
- **Increasing Intervals**: The function is increasing when \( \frac{dy}{dx} > 0 \). Solving the inequality \( 5 - y > 0 \), we find that the function is increasing when \( y < 5 \). This means that for values of \( y \) below 5, as \( x \) increases, so does \( y \).
- **Decreasing Intervals**: Conversely, the function is decreasing when \( \frac{dy}{dx} < 0 \). By solving \( 5 - y < 0 \), we determine that the function decreases in intervals where \( y > 5 \). Here, \( y \) decreases as \( x \) increases.
Understanding these intervals helps in predicting the movement and behavior of the solution in relation to \( x \).

Inflection Points

Inflection points are values at which the concavity of a curve changes. Normally, these points are identified by finding where the second derivative equals zero. Here, we start with \( \frac{dy}{dx} = 5 - y \) and find the second derivative.
By differentiating again with respect to \( x \), we obtain \( \frac{d^2y}{dx^2} = -\frac{dy}{dx} \). For an inflection point, \( \frac{d^2y}{dx^2} = 0 \), leading to \(-\frac{dy}{dx} = 0 \). Solving \( \frac{dy}{dx} = 0 \), which gives \( 5 - y = 0 \), shows that \( y = 5 \).
Instead of indicating an inflection point, this serves as a point of balance where the rates of increase and decrease cancel each other. It indicates the stability of the solution at \( y = 5 \).

Solution Curve Sketching

Sketching the solution curve provides a visual understanding of the differential equation's behavior. For the equation \( \frac{dy}{dx} = 5 - y \):

• Start with the constant solution \( y = 5 \), which is a horizontal line across the graph indicating no change in \( y \) irrespective of the \( x \)-values.
• For initial values of \( y < 5 \), the curve starts below this line and is defined as increasing, as seen from the increasing intervals.
• On the other hand, for initial values of \( y > 5 \), the curve starts above the line, decreasing as \( x \) grows, which aligns with the decreasing intervals.
• Both sets of curves approach the horizontal line \( y = 5 \) asymptotically, meaning they get closer to this line but never touch or cross it as \( x \) approaches infinity.

This sketching technique reveals the overall dynamic behavior of potential solutions of the differential equation, giving clearer insight into their long-term tendencies.