Problem 24 - Calculus Early Transcendentals

Question

(a) Find a slope field whose integral curve through \((0,0)\) satisfies \(x e^{y}+y e^{x}=0\) by differentiating this equation implicitly. (b) Prove that if \(y(x)\) is any integral curve of the slope field in part (a), then \(x e^{y(x)}+y(x) e^{x}\) will be a constant function. (c) Find an equation that implicitly defines the integral curve through \((1,1)\) of the slope field in part (a).

Step-by-Step Solution

Answer

(a) \(\frac{dy}{dx} = \frac{-e^{y} - y e^{x}}{x e^{y} + e^{x}}\); (b) \(x e^{y} + y e^{x}\) is constant; (c) \(x e^{y} + y e^{x} = 2e\).

Step 1: Differentiate implicitly

Starting with the given equation \(x e^{y} + y e^{x} = 0\), differentiate both sides with respect to \(x\). We use the product rule on both terms: \(d/dx[ x e^{y} ] = e^{y} + x e^{y} \frac{dy}{dx}\) and \(d/dx[ y e^{x} ] = e^{x} \frac{dy}{dx} + y e^{x}\). Setting the derivative to zero because the derivative of zero is zero, we have: \[ e^{y} + x e^{y} \frac{dy}{dx} + e^{x} \frac{dy}{dx} + y e^{x} = 0 \]

Step 2: Solve for \(\frac{dy}{dx}\)

Rearrange the terms obtained in the derivative to solve for \(\frac{dy}{dx}\):\[ x e^{y} \frac{dy}{dx} + e^{x} \frac{dy}{dx} = - e^{y} - y e^{x} \]Factoring out \(\frac{dy}{dx}\) from the left-hand side gives:\[ \frac{dy}{dx} (x e^{y} + e^{x}) = - e^{y} - y e^{x} \]Thus, solving for \(\frac{dy}{dx}\) gives: \[ \frac{dy}{dx} = \frac{-e^{y} - y e^{x}}{x e^{y} + e^{x}} \]

Step 3: Verify it's a constant function

To confirm that \(x e^{y(x)} + y(x) e^{x}\) is a constant function when \(y(x)\) is an integral curve, differentiate \(x e^{y} + y e^{x}\) to check if the derivative is zero. From the differentiation earlier in Step 1, all terms sum to zero:\[ e^{y} + x e^{y} \frac{dy}{dx} + e^{x} \frac{dy}{dx} + y e^{x} = 0 \]This confirms the expression remains constant along the integral curves.

Step 4: Determine the integral curve through (1,1)

Setting the expression \(x e^{y} + y e^{x} = C\) and using the point \((1,1)\), substitute these values into the expression:\[ 1 \cdot e^{1} + 1 \cdot e^{1} = C \]Simplifying gives \(2e = C\). Thus, the integral curve through \((1,1)\) is implicit in the equation: \[ x e^{y} + y e^{x} = 2e \]

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used in calculus to find the derivative of a function when it is not explicitly solved for one variable in terms of another. In simple terms, when you have an equation involving both variables, implicit differentiation helps you differentiate each variable separately.

In the exercise given, you have the equation \(xe^{y} + ye^{x} = 0\). This equation isn't solved explicitly for \(y\) in terms of \(x\), so implicit differentiation comes in handy. Here's what happens step by step:

• Differentiate each term separately. This involves applying the product rule, as each term is a product of functions of \(x\) and \(y\).
• For \(xe^{y}\), differentiate as: \(e^{y} + xe^{y}\frac{dy}{dx}\).
• For \(ye^{x}\), the differentiation becomes: \(e^{x}\frac{dy}{dx} + ye^{x}\).
• Since you are differentiating both sides of a zero equation, all terms must add up to zero after multiplication and rearrangement.

This process reveals the slope of the tangent line to the curve at any given point, shown as \(\frac{dy}{dx}\). It's crucial because it allows you to describe how the curve behaves even when \(y\) isn't isolated. This highlights the power of implicit differentiation in analyzing complex relationships between variables.

Integral Curve

An integral curve represents a solution to a differential equation system within a slope field, tracing a path through it. Think of it as a visual translation of algebraic expressions into the pattern or arrangement of curves in a graph.

In this context, given the differential equation derived from implicit differentiation: \(\frac{dy}{dx} = \frac{-e^{y} - ye^{x}}{xe^{y} + e^{x}}\), each solution \(y(x)\) to this differential equation aligns with an integral curve. These curves are integral because they encompass every solution of \(y'(x)\) that satisfies the original condition or equation.

• Integral curves give insights into the long-term behavior of solutions of a system of differential equations, essentially describing how particular solutions evolve.
• The specific function or path these curves follow satisfies the differential equation at each point while passing through a specific initial point, such as \((0,0)\) or \((1,1)\) in the exercise.
• Every point on an integral curve provides essential data about the solution consistency with the overall function's constraints.

In simpler terms, integral curves not only describe movement along paths defined by differential equations but also show how these paths converge or diverge, resembling a system's dynamic process.

Constant Function

A constant function is a function whose value remains the same regardless of its input. In calculus, when we refer to something being a constant, it often relates to values that remain unchanged within certain conditions or constraints of a system.

In the given problem, you have proven that the expression \(xe^{y(x)} + y(x)e^{x}\) remains constant for any integral curve of the derived slope field. This means that once an integral curve satisfies this condition, the same condition holds true along every point on that curve.

• When differentiating \(xe^{y(x)} + y(x)e^{x}\), if all resulting terms sum to zero, it signifies that the derivative of this expression is zero.
• A zero derivative suggests that the function does not change as \(x\) changes, which confirms its constancy.
• This constant function essentially describes conserved quantities along integral curves, indicating invariant properties throughout the dynamics of the system.

This concept implies profound stability in the behavior of solutions across the differential equation landscape, telling us that some properties of the system remain steadfast regardless of the variables it might encounter along its path.