Problem 35 - Calculus

Question

Give an example of: A differential equation that has a logarithmic function as a solution.

Step-by-Step Solution

Answer

An example is the differential equation \( \frac{dy}{dx} = \frac{1}{x} \).

Step 1: Understand the Problem

We need to find a differential equation for which the solution is a logarithmic function like \( \ln(x) \). This involves not only knowing the form of the logarithmic function but also being able to differentiate it and form a differential equation.

Step 2: Differentiate the Logarithmic Function

Consider the function \( y = \ln(x) \). The first derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{dy}{dx} = \frac{1}{x} \). This derivative will play a crucial role in forming our differential equation.

Step 3: Formulate the Equation

Using the derivative \( \frac{dy}{dx} = \frac{1}{x} \), we can construct the differential equation \( \frac{dy}{dx} = \frac{1}{x} \). This equation suggests that the change in \( y \) with respect to \( x \) is inversely proportional to \( x \), which corresponds to a logarithmic relationship.

Step 4: Verify the Solution

To check that \( y = \ln(x) \) satisfies \( \frac{dy}{dx} = \frac{1}{x} \), substitute \( y \) back into the equation. Since we have \( \frac{dy}{dx} = \frac{1}{x} = \frac{1}{x} \), the solution is verified, confirming \( y = \ln(x) \) is indeed a solution to the differential equation.

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that include the logarithm of a variable or number. The natural logarithm, denoted as \( \ln(x) \), is one of the most commonly used logarithmic functions. It represents the power to which the base \( e \) (approximately 2.718) must be raised to yield the number \( x \).

Understanding logarithmic functions is key in numerous mathematical calculations such as solving exponential equations, integrating functions, and modeling real-world phenomena like population growth or radioactive decay. Logarithms help simplify complex multiplicative relationships into additive ones, making calculations easier.

• Standard form: The function \( y = \ln(x) \) expresses \( y \) as the logarithm of \( x \).
• Domain and range: For \( \ln(x) \), the domain is \( x > 0 \), and the range is all real numbers.
• Properties: Logarithms convert multiplication into addition: \( \ln(ab) = \ln(a) + \ln(b) \).

Derivative

The derivative is a fundamental concept in calculus, representing the rate of change of a function with respect to a variable. For a logarithmic function like \( y = \ln(x) \), its derivative tells us how \( y \) changes as \( x \) changes.

The process of differentiation involves finding this rate of change. For \( \ln(x) \), the derivative is \( \frac{dy}{dx} = \frac{1}{x} \), which signifies that the slope of the tangent to the curve at any point \( x \) is the reciprocal of \( x \).

• Meaning: Derivatives explain how a function's output is affected by slight changes in input.
• Calculation: Differentiate \( y = \ln(x) \) to get \( \frac{dy}{dx} = \frac{1}{x} \).
• Applications: Differentiation is used for finding minimum and maximum values, understanding motion in physics, and more.

First Order Differential Equation

A first-order differential equation is an equation that involves the first derivative of a function. In this case, the focus is on equations where the highest derivative is the first one. An example is \( \frac{dy}{dx} = \frac{1}{x} \), a simple yet illustrative equation of this type.

This equation indicates that the rate of change of \( y \) is inversely proportional to \( x \), a hallmark of logarithmic functions. Solving such equations typically involves integrating both sides to find \( y \) in terms of \( x \).

• Characteristics: First-order, involving \( \frac{dy}{dx} \).
• Solutions: Often require integration to solve.
• Utility: Used in fields like physics, engineering, and economics to describe rates of change in systems.

Solution Verification

Verifying a solution to a differential equation involves checking whether a proposed function satisfies the equation when substituted back into it. For \( y = \ln(x) \) and the differential equation \( \frac{dy}{dx} = \frac{1}{x} \), verification is straightforward.

By differentiating \( y = \ln(x) \), we find \( \frac{dy}{dx} = \frac{1}{x} \). Replacing \( y \) and \( \frac{dy}{dx} \) into the original equation confirms the function as a solution:

• Step 1: Differentiate the function to obtain \( \frac{dy}{dx} \).
• Step 2: Substitute \( \frac{dy}{dx} \) and \( y \) into the differential equation.
• Step 3: Check for consistency, ensuring both sides of the equation match.

This process confirms the accuracy of the solution and enhances understanding of how differential equations model real-world phenomena.