Problem 55 - 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 \(d y / d x=e^{-x^{2}}\). (a) Explain why a solution of the DE must be an increasing function on any interval of the \(x\) -axis. (b) What are \(\lim _{x} d y / d x\) and \(\lim d y / d x ?\) What does this \(x \rightarrow \infty\) suggest about a solution curve as \(x \rightarrow \pm \infty\) ? (c) Determine an interval over which solution curve is concave down and an interval over which the curve is concave up. (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

The solution is increasing on all intervals, approaches a horizontal asymptote as \( x \to \pm\infty \), is concave up on \((-\infty, 0)\), and concave down on \((0, \infty)\).

Step 1: Identify nature of function from first derivative

We are given \( \frac{dy}{dx} = e^{-x^2} \). Since \( e^{-x^2} \) is always positive for any real \( x \), the solution \( y = \phi(x) \) must be an increasing function on any interval of the \( x \)-axis.

Step 2: Evaluate limits of the first derivative

Examine \( \lim_{x \to \pm \infty} \frac{dy}{dx} = \lim_{x \to \pm \infty} e^{-x^2} \). Since \( x^2 \to \infty \) as \( x \to \pm \infty \), it follows that \( e^{-x^2} \to 0 \). Thus, \( \lim_{x \to \pm \infty} \frac{dy}{dx} = 0 \), suggesting the solution curve \( y = \phi(x) \) approaches a horizontal asymptote as \( x \to \pm\infty \).

Step 3: Determine concavity with second derivative

To find concavity, calculate the second derivative: \( \frac{d^2y}{dx^2} = -2xe^{-x^2} \). The sign of \( \frac{d^2y}{dx^2} \) tells us about concavity: - For \( x > 0 \), \( \frac{d^2y}{dx^2} < 0 \) (concave down). - For \( x < 0 \), \( \frac{d^2y}{dx^2} > 0 \) (concave up). Thus, the curve is concave up on \( (-\infty, 0) \) and concave down on \( (0, \infty) \).

Step 4: Sketch the solution graph

Given parts (a)-(c), draw a graph: start below a horizontal asymptote, increase across the \( x \)-axis, passing through the point of inflection at \( x=0 \) where the concavity changes, and gradually level off towards the horizontal asymptote as \( x \) approaches \( \pm\infty \).

Key Concepts

First Derivative

The first derivative of a function, denoted as \( \frac{dy}{dx} \), is a fundamental concept in calculus that illustrates the rate of change or the slope of the function at any given point. In the context of our differential equation \( \frac{dy}{dx} = e^{-x^2} \), the first derivative tells us how the function \( y = \phi(x) \) behaves as \( x \) changes.

Since the exponential function \( e^{-x^2} \) is always positive for any real \( x \), this indicates that the solution \( y = \phi(x) \) is consistently increasing over every interval of the \( x \)-axis. An increasing function implies that as \( x \) grows, \( y \) either rises or stays steady but never decreases, aligning with our interpretation that the solution is an upward-moving curve.

- **Importance:** Helps in predicting whether the solution function is increasing or decreasing.- **Outcome:** Knowing \( \frac{dy}{dx} = e^{-x^2} \) is always positive tells us that the function never declines, always climbs.

Second Derivative

The second derivative, \( \frac{d^2y}{dx^2} \), offers insights into the curvature or concavity of the function \( y = \phi(x) \). By differentiating \( \frac{dy}{dx} = e^{-x^2} \) again, we obtain \( \frac{d^2y}{dx^2} = -2xe^{-x^2} \).

This second derivative has a sign that depends on the value of \( x \):

• For \( x > 0 \), \( \frac{d^2y}{dx^2} \, < \, 0 \), indicating the function is concave down.
• For \( x < 0 \), \( \frac{d^2y}{dx^2} \, > \, 0 \), indicating concavity up.

This change in sign at \( x = 0 \) informs us that the function has an inflection point here, where the concavity switches. The presence of the second derivative reveals much about how the function curves and shifts, aiding in sketching an accurate graph.

- **Sign:** Affects whether the curve bends upward or downward.- **Transition:** Second derivative zero or sign change suggests an inflection point.

Concavity

Concavity describes the direction that a curve bends. It's determined by the second derivative, \( \frac{d^2y}{dx^2} \). When \( \frac{d^2y}{dx^2} \) is positive, the function is concave up, resembling a cup that holds water. Conversely, when \( \frac{d^2y}{dx^2} \) is negative, the function is concave down, like an upside down cup.

For our specific function:

• The curve is concave up over \( (-\infty, 0) \).
• The curve is concave down on \( (0, \infty) \).

Understanding concavity allows for more accurate graph construction and prediction of the function’s behavior between different intervals. This transition at \( x = 0 \) is key in identifying the inflection point, where curvature changes.

- **Upward vs Downward:** Concavity up means the function is shaped upward; concavity down means it bends downward.- **Graphical Interpretation:** Crucial for plotting the right structure and curve of the graph.

Limit Behavior

Limit behavior explores what happens to the function \( y = \phi(x) \) as \( x \) approaches very large or very small values, i.e., \( x \to \pm \infty \). For \( \frac{dy}{dx} = e^{-x^2} \), as \( x^2 \) becomes infinite, \( e^{-x^2} \) tends to 0.

This limit finding means that the rate of increase of \( y \) diminishes as \( x \) extends toward infinity or negative infinity. Both ends approaching zero suggests a horizontal asymptote in the solution curve. Thus, the function \( y = \phi(x) \) rises and approximately plateaus as \( x \) goes to \( \pm \infty \).

- **Approach:** Asymptotically the tangent approaches zero, meaning horizontal flattening.- **Outcome:** Curve stabilizes, without further sharp increases or decreases.