Problem 50 - Calculus - AP Edition

Question

Verify that \(y=-\ln \left(e^{2}-x\right)\) satisfies \(d y / d x=e^{y},\) with \(y=-2\) when \(x=0\)

Step-by-Step Solution

Answer

Yes, the function satisfies \( dy/dx = e^y \) and the initial condition is met.

Step 1: Find the derivative of y with respect to x

Given the function \( y = -\ln(e^2 - x) \), we need to differentiate it with respect to \( x \). Using the chain rule, the derivative of \( y \) is \( dy/dx = -\frac{1}{e^2 - x} \times (-1) = \frac{1}{e^2 - x} \).

Step 2: Substitute y into e^y and check the equality

Substitute \( y = -\ln(e^2 - x) \) into \( e^y \) to verify \( dy/dx = e^y \). We have \( e^y = e^{-\ln(e^2-x)} \), which simplifies to \( \frac{1}{e^{\ln(e^2-x)}} = \frac{1}{e^2-x} \).

Step 3: Verify the equality dy/dx = e^y

From Step 1, \( \frac{dy}{dx} = \frac{1}{e^2 - x} \). From Step 2, \( e^y = \frac{1}{e^2-x} \). Since both expressions are equal, \( \frac{dy}{dx} = e^y \), verifying the condition is met.

Step 4: Verify initial condition y=-2 when x=0

Substitute \( x = 0 \) into the original function to verify the initial condition. \( y = -\ln(e^2 - 0) = -\ln(e^2) = -2 \). The initial condition \( y = -2 \) when \( x = 0 \) is satisfied.

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It helps us understand how systems evolve over time and rates at which these changes occur. In this exercise, we delve into differential calculus, which focuses on the concept of taking derivatives. Derivatives are essential for finding the rate of change of a variable in relation to another variable. This forms the core of many scientific and engineering problems.

When working with functions like \( y = -\ln(e^2 - x) \), finding the derivative \( \frac{dy}{dx} \) allows us to understand how \( y \) changes as \( x \) changes. Here, we specifically compute the derivative to ensure it equals a given function, \( e^y \). Key tools from calculus that facilitate this process include rules like the chain rule (which we will discuss shortly), which are fundamental in managing complex operations. This structural approach not only broadens understanding but also simplifies solving these equations.

Chain Rule

The chain rule is a core principle in calculus used to differentiate composite functions. A composite function is essentially a function within another function. For example, if you have a function \( y = -\ln(e^2 - x) \), this is composed of an outer function, \(-\ln(u)\), and an inner function, \(u = e^2 - x\).

• The derivative of the outer function in terms of the inner function, is \(-1/u\).
• The derivative of the inner function \( u = e^2 - x \) is \(-1\).

Multiplying these, as the chain rule prescribes, yields a combined derivative: \(-(-1/u) = \frac{1}{e^2 - x}\). This shows how the chain rule streamlines the process of finding derivatives for complex compositions.

Using the chain rule simplifies tackling complex functions by breaking them into manageable components, ensuring accuracy in differentiation tasks. This aids in effortlessly verifying equations like \( \frac{dy}{dx} = e^y \) by ensuring all parts of a composite are correctly addressed.

Initial Conditions

Initial conditions provide specific values at the outset to verify the solutions of differential equations. These conditions illustrate a point on the function that the solution must satisfy to be valid within the problem's context. In our exercise, the initial condition states \( y = -2 \) when \( x = 0 \).

To verify this, we substitute \( x = 0 \) into the original function. When \( y = -\ln(e^2 - 0) = -\ln(e^2) \), simplifying provides \( y = -2 \). This matches our initial condition, confirming that the solution satisfies both the differential equation and the specific starting values. Using initial conditions like this helps in narrowing down solutions to those that are physically meaningful and consistent with given data.

• Initial conditions are crucial for confirming the relevance and accuracy of general solutions.
• They ensure the solution adheres to the specific scenario posed by the exercise or real-world problem.

Thus, initial conditions play a pivotal role in practical applications where precise outcomes are necessary.