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Chapter 6: Problem 43

Explain what is wrong with the statement. A differential equation cannot have a constant solution.

Short Answer

Expert verified
The statement is incorrect; a differential equation can have a constant solution if the equation is in the form of zero derivatives.

Step by step solution

01

Understanding Differential Equations

A differential equation is an equation that relates a function with its derivatives. It essentially describes how a function changes with respect to its variables.
02

Identifying Constant Solutions

A constant solution in the context of differential equations is a solution where the function does not change, i.e., it remains constant. This means that its derivative is zero, since a constant does not change.
03

Considering the Derivative of a Constant Function

If a function \( y(x) = c \) is constant, its derivative \( \frac{dy}{dx} \) is 0. Consequently, any differential equation of the form \( \frac{dy}{dx} = 0 \) is satisfied by \( y(x) = c \), where \( c \) is any constant.
04

Example of a Differential Equation with Constant Solution

Consider the differential equation \( \frac{dy}{dx} = 0 \). The solution to this equation is \( y(x) = c \) where \( c \) is a constant. This shows that a differential equation can indeed have a constant solution.
05

Conclusion on the Statement

The statement that 'a differential equation cannot have a constant solution' is incorrect. A differential equation can have a constant solution if it is in the correct form, such as \( \frac{dy}{dx} = 0 \).

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Key Concepts

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

Constant Solution
A differential equation can indeed have a constant solution. This might seem surprising at first, because we often think of differential equations as representing change. However, in the case of a constant solution, the function itself remains the same across all its domain; it doesn't vary with respect to its variable.
  • This happens when the derivative of the function is zero everywhere.
  • In mathematical terms, if you have a function defined as y(x) = c, where c is a constant, the derivative will be dy/dx = 0.
This shows that some differential equations are perfectly satisfied by constant solutions. An equation of the form dy/dx = 0 explicitly describes such a situation and assures us that constant solutions do exist for certain forms of differential equations.
Derivatives
Derivatives are at the heart of differential equations. They express how a function changes as its input changes. For example, if you have a function y(x), the derivative dy/dx tells you the rate at which y changes with respect to x.
  • When the derivative of a function is zero, this indicates that there is no change in the function's value鈥攊t is constant.
  • Finding the derivative helps us understand the nature of the function, whether it is increasing, decreasing, or remaining constant.
Thus, derivatives are essential in determining whether a differential equation, such as dy/dx = 0, has a constant solution. They signal when a function has reached steady-state or equilibrium, remaining unchanged.
Function and Derivative Relationship
The relationship between a function and its derivative is key in understanding differential equations. A derivative provides crucial insights into the behavior of the original function.
  • If a function has a derivative that is zero, the original function is a constant, indicating a state of no change.
  • The derivative gives us a tool to analyze how a function behaves, telling us whether it is stable (does not change) or dynamic (changes over time).
In the context of differential equations, particularly dy/dx = 0, this relationship confirms that the function y is indeed a constant across its domain. This deep interplay between the function and its derivative allows mathematicians to predict and describe the behavior of various systems modeled by differential equations.

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Most popular questions from this chapter

(a) Find and graph the general solution of the differential equation $$\frac{d y}{d x}=\sin x+2$$. (b) Find the solution of the initial value problem $$\frac{d y}{d x}=\sin x+2, \quad y(3)=5$$.

Explain what is wrong with the statement. $$\frac{d}{d x} \int_{0}^{5} t^{2} d t=x^{2}$$

Are the statements true or false? Give an explanation for your answer. If \(F(x)\) is an antiderivative of \(f(x),\) then \(y=F(x)\) is a solution to the differential equation \(d y / d x=f(x)\).

Give an example of: Two different functions \(F(x)\) and \(G(x)\) that have the same derivative.

Explain what is wrong with the statement. $$\text { For all } n, \int x^{n} d x=\frac{x^{n+1}}{n+1}+C$$
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