Problem 74 - Calculus and its applications

Question

What function is also its own derivative? Write a differential equation for which this function is a solution. Are there any other solutions to this differential equation? Why or why not?

Step-by-Step Solution

Answer

The function is \( e^x \) and any constant multiple \( Ce^x \) also solves the differential equation \( \frac{dy}{dx} = y \). There are infinite solutions based on the constant \( C \).

Step 1: Identify the problem

We need to find a function that is its own derivative. This means if the function is \( f(x) \), we are looking for \( f'(x) = f(x) \).

Step 2: Recognize the function

The exponential function \( f(x) = e^x \) is the function whose derivative is itself, meaning \( \frac{d}{dx} (e^x) = e^x \).

Step 3: Write the differential equation

The differential equation where the solution is the function that is its own derivative is \( \frac{dy}{dx} = y \).

Step 4: Solve the differential equation

This is a simple differential equation with the solution \( y = Ce^x \), where \( C \) is a constant. The reason \( C \) is included is because it is a constant of integration that accounts for any initial conditions.

Step 5: Discuss uniqueness

The differential equation \( \frac{dy}{dx} = y \) has an infinite number of solutions depending on the initial condition, all in the form \( y = Ce^x \). Thus, \( e^x \) is a solution when \( C = 1 \), but generally, any constant multiplied by \( e^x \) is also a solution.

Key Concepts

Exponential Functions

Exponential functions are a class of mathematical functions characterized by an enormously rapid rate of growth. A typical exponential function takes the form \( f(x) = a^x \), where \( a \) is a constant. One of the most significant exponential functions is \( f(x) = e^x \), where \( e \) is approximately 2.718. This specific function is unique because it is its own derivative.
A derivative represents the rate at which a function changes at any given point, and for \( e^x \), the rate of change equals the value of the function itself at that point. This special property makes \( e^x \) pivotal for solving differential equations where a function's rate of change is proportional to its current value.
In real-world scenarios, exponential functions model many natural processes, including population growth, radioactive decay, and compounded interest, where changes occur at rates proportional to the current state.

Derivative

The concept of the derivative is fundamental in calculus, measuring how a function changes as its input changes. It gives us the slope of the function at any given point and is crucial for understanding the dynamics of changing systems.
For example, with the function \( f(x) = e^x \), its derivative \( f'(x) \) gives the same \( e^x \), indicating that the rate of increase of the function is constant, matching its current value. This property makes exponential functions key in solving differential equations, whereby growth rates must often align directly with function values.

• The derivative of a constant function is zero.
• The power rule allows us to find derivatives of functions like \( x^n \).
• The rule for products (product rule) and quotients (quotient rule) extends basic derivatives to complex expressions.

In our exercise, the derivative \( \frac{dy}{dx} = y \) results in an equation whose solutions depict a family of exponential functions, each scaling according to their own initial conditions.

Initial Conditions

Initial conditions serve as vital parameters in differential equations, determining the specific solution within a family of solutions. When dealing with differential equations, you often encounter expressions such as \( y = Ce^x \).
Here, \( C \) is the constant determined by the initial condition(s) provided. If no initial condition is specified, \( C \) could be any constant, meaning the solution could be any member of an infinite family of functions.
For example, if you're given an initial condition \( y(0) = 1 \), you substitute into the equation to find \( C \) by solving \( 1 = Ce^0 \), which simplifies to \( C = 1 \).

• Initial conditions are necessary to uniquely determine one specific solution out of many possibilities.
• They are particularly important in physical systems to match the function to initial measurements or states.
• Without initial conditions, a differential equation solution remains non-unique, expressing general behaviors but not specific cases.

In essence, initial conditions help narrow down and define the precise trajectory of solutions in mathematical models of real-life phenomena.