Question

Give an example of a function \(f(x)\) where \(f^{\prime}(x)=f(x)\).

Short answer

An example is the exponential function: \( f(x) = e^x \).

Step-by-step solution

Identify the Type of Differential Equation

We are given the equation \( f'(x) = f(x) \). This is a first-order, linear homogeneous differential equation, which means the solution is a function whose derivative is equal to itself.

Recall the Exponential Function Identity

One well-known function that satisfies this condition is the exponential function. The derivative of the exponential function \( e^x \) with respect to \( x \) is \( e^x \) itself. Thus, \( e^x \) is a candidate for \( f(x) \).

Verify the Solution

To ensure that \( f(x) = e^x \) is indeed a solution, differentiate \( f(x) = e^x \) with respect to \( x \). The derivative \( \frac{d}{dx}e^x = e^x \), which matches the original equation \( f'(x) = f(x) \).

Generalize the Solution

The exponential function solution can be broadened to include \( f(x) = Ce^x \) where \( C \) is a constant. Differentiating \( Ce^x \) gives \( C \cdot e^x \), which still satisfies \( f'(x) = f(x) \).

Key concepts

Exponential Function

An exponential function, in its simplest form, is expressed as \( e^x \). Exponential functions like this are unique due to their characteristic growth behavior. The derivative of an exponential function, \( e^x \), is quite special: it returns to itself upon differentiation.- For example, if you take the derivative of \( e^x \), you will end up with \( e^x \) again.This gives exponential functions a unique property of self-similarity. They are particularly useful in modeling growth processes, such as population increase and radioactive decay, where growth rates are proportional to the size of the population.

First-Order Linear Differential Equation

First-order linear differential equations are equations that involve the first derivative of a function, \( f'(x) \), and the function itself, \( f(x) \). These equations take the general form \( f'(x) + P(x)f(x) = Q(x) \), where \( P(x) \) and \( Q(x) \) are given functions. If \( Q(x) = 0 \), the equation is said to be homogeneous, which is the case in our example.- For the equation \( f'(x) = f(x) \), the function \( P(x) = -1 \) and \( Q(x) = 0 \), making it both linear and homogeneous.Such equations are solved using techniques that include separation of variables and integrating factors, depending on the structure of \( P(x) \) and \( Q(x) \). In the given example, the technique simply involved recognizing the form and identifying the exponential function solution.

Homogeneous Equations

Homogeneous equations are a type of differential equation where all the terms involve the function or its derivatives, without any standalone function of the independent variable. In other words, they look like \( f'(x) + P(x)f(x) = 0 \).These equations are pivotal in mathematical modeling as they often describe:- Systems in equilibrium - Natural processes with no external influenceIn the case of the equation \( f'(x) = f(x) \), it is a first-order homogeneous differential equation. The solution set consists of solutions like \( f(x) = Ce^x \), where \( C \) is an arbitrary constant, forming a family of curves rather than just a single function.