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Chapter 11: Problem 44

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. If \(g(x)\) is a different solution to the differential equation \(d y / d x=2 x-y,\) then \(\lim _{x \rightarrow \infty}(g(x)-f(x))=0\) [Hint: Show that \(w=g(x)-f(x)\) satisfies the differential equation \(d w / d x=-w .]\)

Short Answer

Expert verified
True, \( \lim_{x \rightarrow \infty} (g(x) - f(x)) = 0 \).

Step by step solution

01

Understand the Problem

We have a differential equation \( \frac{dy}{dx} = 2x - y \). We assume \( y = f(x) \) is a solution to this equation. We need to determine if a different solution \( g(x) \) to the equation satisfies \( \lim_{x \rightarrow \infty} (g(x) - f(x)) = 0 \).
02

Define the Difference Function \(w\)

Let \( w(x) = g(x) - f(x) \). We are analyzing \( g(x) \) and \( f(x) \) in terms of their difference \( w(x) \). We need to find the behavior of \( w(x) \) as \( x \to \infty \).
03

Find the Differential Equation for \(w(x)\)

Differentiate \( w(x) \) with respect to \( x \): \[ \frac{d}{dx}[g(x) - f(x)] = \frac{dg}{dx} - \frac{df}{dx} \]Substituting the original differential equation for both \( g(x) \) and \( f(x) \):\[ \frac{dg}{dx} = 2x - g(x), \quad \frac{df}{dx} = 2x - f(x) \]Thus,\[ \frac{dw}{dx} = (2x - g(x)) - (2x - f(x)) = -w(x) \]
04

Solve the Differential Equation for \(w(x)\)

The differential equation \( \frac{dw}{dx} = -w(x) \) is separable. We rearrange it as\[ \frac{dw}{w} = -dx \]Integrate both sides: \[ \int \frac{1}{w} dw = \int -1 dx \] This gives \[ \ln |w| = -x + C \] where \( C \) is the constant of integration. Exponentiating both sides, we obtain \[ |w(x)| = e^{-x + C} = Ce^{-x} \] Thus, \[ w(x) = Ce^{-x} \] where \( C \) is the constant.
05

Evaluate \(\lim_{x \rightarrow \infty} w(x)\)

As \( x \to \infty \), the exponential term \( e^{-x} \to 0 \). Therefore, \[ \lim_{x \rightarrow \infty} w(x) = \lim_{x \rightarrow \infty} Ce^{-x} = 0 \] which means \[ \lim_{x \rightarrow \infty} (g(x) - f(x)) = 0 \].

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Key Concepts

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

Limit of a Function
The concept of limits is fundamental in calculus and helps us understand the behavior of functions as they approach a certain point, or as they tend to infinity. In our exercise, we are considering whether the difference between two solutions to a differential equation tends to zero as the independent variable approaches infinity.

This involves examining the limit \(\lim_{x \rightarrow \infty}(g(x) - f(x)) = 0.\)

In simple terms, as we let \(x\) increase indefinitely, we are interested in what happens to the expression \(g(x) - f(x)\). If it approaches zero, it suggests that the solutions \(g(x)\) and \(f(x)\) become increasingly similar as \(x\) grows.
  • The behavior at infinity is crucial in many applications, for instance in physics and engineering, providing insights into asymptotic behavior.
  • Finding the limit often involves simplifying expressions and identifying dominant terms at large values of \(x\).
Separable Differential Equations
Separable differential equations are a special class that can be easily solved using a straightforward technique. Once an equation is identified as such, it can be manipulated into two expressions that involve only one variable each on the different sides of the equation.

Take the equation: \(\frac{dw}{dx} = -w.\)

Here, the variables \(w\) and \(x\) can be separated:
  • Move all terms involving \(w\) to one side: \(\frac{1}{w} dw = -dx.\)
  • Integrate both sides: \(\int \frac{1}{w} dw = \int -1 dx.\)
The separability allows for such manipulation and integration, showing the power of this method. It results in a general solution which includes an arbitrary constant, often derived from an initial condition.
  • It simplifies complex problems and is applicable in various scientific fields.
  • The technique is a foundation for solving more advanced differential equations.
Exponential Functions
Exponential functions occur frequently in mathematics, characterized by the constant \(e\), approximately 2.718, raised to the power of a variable. They exhibit unique growth or decay properties, depending on whether the exponent is positive or negative.

In our solution, the function \(|w(x)| = Ce^{-x}\)utilizes an exponential function to describe the decay of \(w\) over time as \(x\) increases. This term \(Ce^{-x}\) approaches zero as \(x\) goes to infinity.
  • Exponential decay, as seen here, is typically very rapid. This is why \(\lim_{x \rightarrow \infty} Ce^{-x} = 0.\)
  • Such functions are pivotal in natural sciences, modeling processes like radioactive decay and population dynamics.
  • The constant \(C\) determines the initial value or condition of the exponential function.

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Most popular questions from this chapter

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. If \(f(a)=b,\) the slope of the graph of \(f\) at \((a, b)\) is \(2 a-b\)

An aquarium pool has volume \(2 \cdot 10^{6}\) liters. The pool initially contains pure fresh water. At \(t=0\) minutes, water containing 10 grams/liter of salt is poured into the pool at a rate of 60 liters/minute. The salt water instantly mixes with the fresh water, and the excess mixture is drained out of the pool at the same rate ( 60 liters/minute). (a) Write a differential equation for \(S(t),\) the mass of salt in the pool at time \(t .\) (b) Solve the differential equation to find \(S(t)\) (c) What happens to \(S(t)\) as \(t \rightarrow \infty ?\)

In Problems \(55-58,\) give an example of: A differential equation that is not separable.

Policy makers are interested in modeling the spread of information through a population. For example, agricultural ministries use models to understand the spread of technical innovations or new seed types through their countries. Two models, based on how the information is spread, follow. Assume the population is of a constant size \(M\) (a) If the information is spread by mass media (TV, radio, newspapers), the rate at which information is spread is believed to be proportional to the number of people not having the information at that time. Write a differential equation for the number of people having the information by time \(t .\) Sketch a solution assuming that no one (except the mass media) has the information initially. (b) If the information is spread by word of mouth, the rate of spread of information is believed to be proportional to the product of the number of people who know and the number who don't. Write a differential equation for the number of people having the information by time \(t .\) Sketch the solution for the cases in which (i) No one \(\quad\) (ii) \(5 \%\) of the population (iii) \(75 \%\) of the population knows initially. In each case, when is the information spreading fastest?

According to an article in The New York Times,' ' pigweed has acquired resistance to the weedkiller Roundup. Let \(N\) be the number of acres, in millions, where Roundup-resistant pigweed is found. Suppose the relative growth rate, \((1 / N) d N / d t,\) was \(15 \%\) when \(N=5\) and \(14.5 \%\) when \(N=10 .\) Assuming the relative growth rate is a linear function of \(N,\) write a differential equation to model \(N\) as a function of time, and predict how many acres will eventually be afflicted before the spread of Roundup-resistant pigweed halts.
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