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Chapter 9: Problem 10

Solve the differential equations. \(x \frac{d y}{d x}=\frac{\cos x}{x}-2 y, \quad x>0\)

Short Answer

Expert verified
The solution is \( y = \frac{\sin x + C}{x^2} \), where \( C \) is a constant.

Step by step solution

01

Identify the Type of Differential Equation

The given differential equation is \( x \frac{dy}{dx} = \frac{\cos x}{x} - 2y \). This is a first-order linear differential equation because it can be expressed in the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \) where \( P(x) = \frac{2}{x} \) and \( Q(x) = \frac{\cos x}{x^2} \).
02

Find the Integrating Factor

The integrating factor \( \mu(x) \) is found by calculating \( e^{\int P(x) \, dx} \). Here, \( P(x) = \frac{2}{x} \), so the integral is \( \int \frac{2}{x} \, dx = 2\ln|x| = \ln x^2 \). Therefore, the integrating factor is \( \mu(x) = e^{\ln x^2} = x^2 \).
03

Multiply the Differential Equation by the Integrating Factor

Multiply the entire differential equation by the integrating factor \( x^2 \): \[ x^2 \frac{dy}{dx} + 2x y = \cos x \]. This step transforms the left side into the derivative of \( x^2 y \).
04

Express as an Exact Derivative

Rewrite the equation as the derivative of a product: \[ \frac{d}{dx}(x^2 y) = \cos x \].
05

Integrate Both Sides

Integrate both sides with respect to \( x \): \[ \int \frac{d}{dx}(x^2 y) \, dx = \int \cos x \, dx \].The left side integrates to \( x^2 y \), and the right side integrates to \( \sin x + C \) where \( C \) is the integration constant.
06

Solve for y

Solve for \( y \): \[ x^2 y = \sin x + C \]. Thus, \( y = \frac{\sin x + C}{x^2} \).

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Key Concepts

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

First-order Linear Differential Equation
In mathematics, a first-order linear differential equation is a type of equation that involves a function and its first derivative. These equations are linear concerning the unknown function and its derivative. They can generally be put into the form:
  • \( \frac{dy}{dx} + P(x)y = Q(x) \)
where \( P(x) \) and \( Q(x) \) are functions solely depending on \( x \).
To solve these equations, one of the common strategies is to make them exact using an integrating factor, transforming the problem into something manageable. Recognizing this form from a broader equation allows mathematicians to apply systematic techniques to arrive at a solution.
Integrating Factor
The concept of an integrating factor is crucial for solving first-order linear differential equations. An integrating factor is a function, typically denoted as \( \mu(x) \), used to multiply the entire differential equation, rendering the left-hand side an exact derivative.
How do we find this \( \mu(x) \)?! The strategy involves computing \( e^{\int P(x) \, dx} \).
For instance, if \( P(x) = \frac{2}{x} \), you compute:
  • \( \int \frac{2}{x} \, dx = 2\ln|x| \)
Hence, \( \mu(x) = e^{\ln x^2} = x^2 \).
Applying the integrating factor transforms the equation into a form that readily enables integration, simplifying the resolution of the differential equation.
Exact Derivative
An essential component of solving differential equations is rewriting or transforming them into a form involving an exact derivative. If a differential equation has already been multiplied by the integrating factor, it typically transforms into:
  • \( \frac{d}{dx}(v(x)y) = f(x) \)
Here, \( v(x) \) is the integrating factor, converting the equation into the derivative of a product of functions.
In the exercise, after multiplying by the integrating factor \( x^2 \), it becomes:
  • \( \frac{d}{dx}(x^2 y) = \cos x \)
This transformation allows for straightforward integration on both sides, simplifying the process significantly into integration of single functions.
Integration Constant
An integral part of solving differential equations is the concept of the integration constant, denoted often by \( C \). This constant arises because indefinite integration produces a family of solutions that differ by a constant.
For instance, integrating \( \int \cos x \, dx \) yields \( \sin x + C \), indicating that there is not just one solution but a continuum of solutions along \( C \).
In solving the given differential equation, after integrating, you're left with:
  • \( x^2 y = \sin x + C \)
This constant \( C \) is critical when solving initial value problems, where conditions are given to define the specific solution of the differential equation out of the myriad of possibilities.

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Most popular questions from this chapter

Solve the Bernoulli equations. \(x^{2} y^{\prime}+2 x y=y^{3}\)

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\). c. Sketch several solution curves. $$\frac{d y}{d x}=y^{3}-y$$

Solve the Bernoulli equations. \(y^{\prime}-y=x y^{2}\)

In 1925 Lotka and Volterra introduced the predator-prey equations, a system of equations that models the populations of two species, one of which preys on the other. Let \(x(t)\) represent the number of rabbits living in a region at time \(t,\) and \(y(t)\) the number of foxes in the same region. As time passes, the number of rabbits increases at a rate proportional to their population, and decreases at a rate proportional to the number of encounters between rabbits and foxes. The foxes, which compete for food, increase in number at a rate proportional to the number of encounters with rabbits but decrease at a rate proportional to the number of foxes. The number of encounters between rabbits and foxes is assumed to be proportional to the product of the two populations. These assumptions lead to the autonomous system $$\begin{aligned}\frac{d x}{d t} &=(a-b y) x \\\\\frac{d y}{d t} &=(-c+d x) y\end{aligned}$$ where \(a, b, c, d\) are positive constants. The values of these constants vary according to the specific situation being modeled. We can study the nature of the population changes without setting these constants to specific values. What happens to the fox population if there are no rabbits present?

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$y^{\prime}=2 x y+2 y, \quad y(0)=3, \quad d x=0.2$$
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