Problem 59 - Calculus - AP Edition

Question

Determine whether the statement is true or false. Explain your answer. If a function \(y=f(x)\) satisfies \(d y / d x=y,\) then \(y=e^{x}\)

Step-by-Step Solution

Answer

True, \( y = e^x \) is a specific solution of the differential equation \( \frac{dy}{dx} = y \).

Step 1: Identify the Given Differential Equation

The problem statement provides the differential equation \( \frac{dy}{dx} = y \). This means the derivative of \( y \) with respect to \( x \) must equal the function itself.

Step 2: Recall the Exponential Function Property

The exponential function \( y = e^x \) has a special property: its derivative with respect to \( x \) is itself, \( \frac{d}{dx}[e^x] = e^x \). This matches the form of the given differential equation.

Step 3: Verify \( y = e^x \) as a Solution

Substitute \( y = e^x \) into the differential equation \( \frac{dy}{dx} = y \). Compute the derivative: \( \frac{dy}{dx} = e^x \). Since \( \frac{dy}{dx} = y \) simplifies to \( e^x = e^x \), the equality holds true. Therefore, \( y = e^x \) satisfies the differential equation.

Step 4: Analyze Completeness

Consider other functions that could satisfy \( \frac{dy}{dx} = y \). Note that any function of the form \( y = Ce^x \), where \( C \) is a constant, will satisfy this differential equation. However, the specific function \( y = e^x \) represents the case where \( C = 1 \). Thus, while not the general solution, \( y = e^x \) is a specific valid solution.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that can be represented as \( y = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithms, approximately equal to 2.71828. These functions are particularly important because they describe growth or decay processes where the rate of change is proportional to the value of the function itself.

In the given exercise, we deal with the simplest form of an exponential function, where \( a = 1 \) and \( b = 1 \). This results in the function \( y = e^x \). What's interesting about \( y = e^x \) is its unique property of having its derivative equal to itself, making it suitable for solving many differential equations involving growth phenomena.

Exponential functions can model diverse real-life processes, such as population growth, radioactive decay, and continuously compounding interest. Understanding their properties helps in solving differential equations like \( \frac{dy}{dx} = y \), where identifying \( y = e^x \) as a solution is key.

Derivative

The derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to a variable. When we say the derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} \), it implies how quickly \( y \) changes as \( x \) changes. It's like the function's speedometer, telling us the function's slope at any given point.

Consider the differential equation \( \frac{dy}{dx} = y \). Here, the derivative of \( y \), which is \( \frac{dy}{dx} \), is equal to \( y \) itself. This indicates a self-proportional rate of change: the growth rate of the function is directly proportional to its current value. This kind of relationship commonly arises in exponential growth scenarios.

For the function \( y = e^x \), the derivative \( \frac{d}{dx}[e^x] = e^x \) embodies this concept perfectly. This self-similar property under differentiation is rare and makes exponential functions especially suitable for modeling specific natural and physical processes.

Function Property

Functions have unique properties that distinguish them from one another. For an exponential function like \( y = e^x \), one key property is that its derivative is the same as the function itself, \( \frac{d}{dx}[e^x] = e^x \). This property is not only intriguing but also useful for solving differential equations.

In the context of the differential equation \( \frac{dy}{dx} = y \), the function \( y = e^x \) is a solution because it satisfies the equation's requirement: the rate of change \( \frac{dy}{dx} \) matches the function \( y \). There are other functions that satisfy this equation, like \( y = Ce^x \), where \( C \) is any constant, showcasing the variety of possible solutions.

The specific solution \( y = e^x \) serves to demonstrate the uniqueness of its property. However, when considering the broader solution set, we see that any constant multiple of \( e^x \) will also work, illustrating the scalability of exponential functions. This further emphasizes the general solution in differential equations, which includes all possible functions that satisfy the given condition.