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Chapter 11: Problem 45

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

Short Answer

Expert verified
An example is \( \frac{dy}{dx} = \frac{1}{x} \), with solution \( y = \ln(x) \).

Step by step solution

01

Choose a Logarithmic Function

Let's consider the logarithmic function as our solution: \[ y = \ln(x) \] This is our candidate solution for the differential equation.
02

Differentiate the Logarithmic Function

Differentiate \( y = \ln(x) \) with respect to \( x \). The derivative of \( \ln(x) \) is:\[ \frac{dy}{dx} = \frac{1}{x} \]
03

Write the Resulting Differential Equation

Plug the derivative back into the differential equation to express it in terms of \( y \): \[ \frac{dy}{dx} = \frac{1}{x} \] This is our differential equation where the solution \( y = \ln(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Function
A logarithmic function is one of the basic functions in mathematics, typically written as \( y = \ln(x) \) for the natural logarithm. This function is defined for all positive values of \( x \), providing outputs that reveal the power to which a base number must be raised to achieve a specific value.

Key properties of the logarithmic function help in solving various mathematical problems:
  • The logarithm of 1 is always zero, i.e., \( \ln(1) = 0 \).
  • Logarithms turn multiplication into addition and division into subtraction, which helps simplify complex calculations.
  • The natural logarithm, \( \ln(x) \), is the inverse function of the exponential function, \( e^x \).
  • A steep climb is observed near zero, which gradually flattens out as \( x \) increases.
Understanding these properties enables us to effectively use logarithmic functions in numerous problems, including solving differential equations.
Solution of Differential Equations
A differential equation involves an unknown function and its derivatives. It is a powerful tool in mathematics to describe various physical phenomena, ranging from motion to electricity to population dynamics.

To solve a differential equation is to find the original function that satisfies the equation. For example, in the exercise, the derivative \( \frac{dy}{dx} = \frac{1}{x} \) suggests that the original function approximates the rate change at any \( x \):
  • Recognize that the differential equation models how \( y \) changes with respect to \( x \).
  • Integrate the equation, reversing differentiation. Here, the primitive function is \( y = \ln(x) \), fitting perfectly because the derivative \( \frac{1}{x} \) aligns with our original differential equation.
  • Verify the solution by differentiating back to see if it matches the differential equation.
Effective solutions require solid integration skills and an understanding of differential behavior.
Differentiation
Differentiation is a fundamental concept in calculus that deals with the rate of change of a function. It involves computing the derivative, determining how a function changes as its input changes.

For the natural logarithm \( \ln(x) \), differentiation is a straightforward process:
  • The derivative of \( \ln(x) \) is \( \frac{1}{x} \). This implies that per-unit increase in \( x \), \( y \) increases by \( \frac{1}{x} \).
  • Differentiation is used to identify slopes of curves at any point. For example, the graph of \( \ln(x) \) becomes less steep as \( x \) increases.
  • The process can handle limits and instantaneous rates, tracking tiny changes too complex for algebraic manipulations alone.
Differentiation is crucial for forming differential equations and finding solutions that fit given functions, especially when doing the reverse process through integration.

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Most popular questions from this chapter

Find the general solution to the given differential equation. $$\frac{d^{2} p}{d t^{2}}+\frac{d p}{d t}+p=0$$

Find the general solution to the given differential equation. $$z^{\prime \prime}+2 z^{\prime}=0$$

Find the general solution to the given differential equation. $$y^{\prime \prime}-3 y^{\prime}+2 y=0$$

Solve the initial value problem. $$y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=1, y^{\prime}(0)=0$$

Explain what is wrong with the statement. The maximum rate of change occurs at \(t=25\) for a quantity \(Q\) growing according to the logistic equation $$\frac{d Q}{d t}=0.13 Q(1-0.02 Q)$$
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