Problem 41 - Advanced Engineering Mathematics

Question

What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple \(k\) of itself? Write each answer in the form of a first-order differential equation with a solution.

Step-by-Step Solution

Answer

The function is \( y = Ce^x \) or \( y = Ce^{kx} \).

Step 1: Identify the problem

We need to find a function whose derivative is the function itself, and another function whose derivative is a constant multiple of itself. These must be written as first-order differential equations and solved.

Step 2: Find the function with its derivative equal to itself

The problem asks for a function that satisfies the equation \( f'(x) = f(x) \). The solution to this differential equation is the exponential function. Specifically, \( f(x) = Ce^x \), where \( C \) is a constant, satisfies this condition because the derivative of \( Ce^x \) with respect to \( x \) is indeed \( Ce^x \).

Step 3: Find the function with its derivative as a constant multiple of itself

We need a function that satisfies \( f'(x) = kf(x) \), where \( k \) is a constant. The exponential function also satisfies this condition but with a different argument; \( f(x) = Ce^{kx} \) is a solution. The derivative of \( Ce^{kx} \) is \( Cke^{kx} \), which simplifies to \( kCe^{kx} = kf(x) \).

Step 4: Write the differential equations and solutions

For the first part, we have the differential equation \( \frac{dy}{dx} = y \) with the solution \( y = Ce^x \). For the second part, the differential equation is \( \frac{dy}{dx} = ky \) with the solution \( y = Ce^{kx} \).

Key Concepts

Exponential Functions

Exponential functions are a special class of functions that exhibit a constant rate of growth or decay. They are defined by expressions such as \( f(x) = Ce^x \), where \( C \) is a constant and \( e \) is the base of the natural logarithm, approximately equal to 2.718.

These functions are unique because their rate of change is directly proportional to their value. This property makes them extremely useful in modeling real-world phenomena, such as population growth, radioactive decay, and interest calculations.

• Exponential functions have the general form: \( f(x) = Ce^{kx} \).
• When \( k = 1 \), the function is simply \( Ce^x \).
• The constant \( C \) affects the function's initial value or height.

The function \( Ce^{kx} \) is versatile in that by adjusting \( k \), we can model various growth rates. If \( k \) is positive, the function represents growth; if negative, it represents decay. The ability to describe such a wide range of behaviors makes exponential functions powerful tools in mathematics and applied sciences.

First-Order Differential Equations

First-order differential equations are equations involving the first derivative of a function. They are fundamental in calculus and appear in various practical applications ranging from physics to engineering.

The most basic form of a first-order differential equation is \( \frac{dy}{dx} = ky \), where \( y \) is a function of \( x \) and \( k \) is a constant. These equations are used to describe situations where the rate of change of a quantity is proportional to the quantity itself, such as exponential growth or decay.

• The differential equation \( \frac{dy}{dx} = y \) describes a function whose rate of change is the same as the function itself.
• For a general constant \( k \), the equation becomes \( \frac{dy}{dx} = ky \).
• Solutions to these equations are typically of the form \( y = Ce^{kx} \).

These equations are straightforward to solve using methods like separation of variables, which involve rearranging terms to isolate the derivative and integrate both sides. Understanding first-order differential equations provides a foundation for more complex differential equations used in advanced calculus and applied mathematics.

Calculus Concepts

Calculus is the mathematical study of continuous change, and it is thoroughly embedded in the analysis of limits, derivatives, and integrals.

Understanding derivatives, such as in the differential equations mentioned, is central to calculus. A derivative represents an instantaneous rate of change or the slope of the tangent line to the graph of a function at any point. This concept underlies much of the manipulation in solving differential equations.

• Derivatives provide information about the function's growth, decay, or concavity.
• They allow us to solve for functions given rates of change, crucial in forming first-order differential equations.
• Finding derivatives involve rule applications such as the power, product, and chain rules.

Solving a differential equation effectively reverses the process of differentiation to find a function given derivatives. This not only involves algebraic manipulation but also an understanding of integration, which is essentially the reverse operation to differentiation, adding another layer to calculus.