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Chapter 9: Problem 35

This last equation is linear in the (unknown) dependent variable u. Solve the differential equations. \(x y^{\prime}+y=y^{-2}\)

Short Answer

Expert verified
The solution is \( y = \left(\frac{|C|}{x}\right)^{1/3} \).

Step by step solution

01

Identify the Equation Type

The given equation is \( x y' + y = y^{-2} \), which is a first-order differential equation because it involves the first derivative of \( y \). We can solve this using an appropriate method for first-order equations.
02

Rearrange Equation

Rearrange the equation to express it in standard linear form. The equation \( x y' + y = y^{-2} \) can be written as \( y' + \frac{1}{x} y = \frac{1}{x y^2} \). This is now in the standard linear form \( y' + P(x)y = Q(x) \) where \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{1}{xy^2} \).
03

Determine Integrating Factor

For a linear first-order differential equation, we find an integrating factor \( \mu(x) \) using \( \mu(x) = e^{\int P(x) \, dx} \). Here, \( P(x) = \frac{1}{x} \), so \( \mu(x) = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \).
04

Solve Using Integrating Factor

Multiply the entire differential equation by the integrating factor \( |x| \):\[ |x| y' + |x|^2 y = \frac{|x|}{x y^2} \]. This simplifies to \( (|x|y)' = \frac{1}{y^2} \).
05

Integrate Both Sides

Integrate both sides with respect to \( x \). The left side becomes \( |x| y = \int \frac{1}{y^2} \, dx \), which becomes \( |x| y = -\frac{1}{y} + C \) after integrating \( -\frac{1}{y} \).
06

Solve for y

Rearrange the integrated equation to solve for \( y \). We obtain \( |x| y^3 = -1 + Cy^3 \). This implies \( y^3 = \frac{|C|}{x} \). Take the cube root to find \( y \): \( y = \left(\frac{|C|}{x}\right)^{1/3} \).

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Key Concepts

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

Integrating Factor
An integrating factor is a crucial component when solving linear first-order differential equations. It transforms these equations into a form that is easier to integrate and solve. To find the integrating factor, denoted as \( \mu(x) \), we use the expression \( \mu(x) = e^{\int P(x) \, dx} \), where \( P(x) \) is the coefficient function of \( y \) in the equation.

In our exercise, the differential equation is already in linear form:
  • \( y' + \frac{1}{x} y = \frac{1}{xy^2} \)
  • The function \( P(x) = \frac{1}{x} \) is identified here.
The next step is to determine the integrating factor:
  • Calculate \( \mu(x) = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \).
This integrating factor \( |x| \) enables us to rewrite the differential equation, making it integrable and leading us to the solution more effectively.
Linear Differential Equation
A linear differential equation is characterized by its order and linearity over the unknown function and its derivatives. In our context, the equation is both a first-order and linear differential equation because it involves only the first derivative of \( y \), without any exponents or multiplications among the derivatives and the solution itself.

To identify a linear differential equation, it follows the general form:
  • \( y' + P(x) y = Q(x) \)
In this exercise, this form is represented as:
  • \( y' + \frac{1}{x} y = \frac{1}{x y^2} \)
  • \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{1}{x y^2} \)
By recognizing and rearranging the equation into this standard form, we're able to apply the method of integrating factors to find a solution efficiently.
Solving Differential Equations
Solving differential equations involves several strategic steps that once mastered, simplify seemingly complicated equations.

For our linear first-order differential equation, after identifying the equation form and determining the integrating factor, the next step is to solve it using the integrating factor:
  • Multiplied through by \( |x| \), the equation becomes \( (|x| y)' = \frac{1}{y^2} \).
  • Integrate both sides with respect to \( x \).
This integration provides a direct path to the general solution. The encoded equation transforms to:
  • Integrate to get \( |x| y = -\frac{1}{y} + C \)
  • Rearrange it to \( |x| y^3 = -1 + Cy^3 \).
The final solution for \( y \) is derived by isolating \( y \):
  • Extract from the cube: \( y = \left(\frac{|C|}{x}\right)^{1/3} \)
This step-by-step approach, along with understanding of linearity and integrating factors, allows us to solve these differential equations systematically.

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Most popular questions from this chapter

Current in an open \(R L\) circuit If the switch is thrown open after the current in an \(R L\) circuit has built up to its steady-state value \(I=V / R,\) the decaying current (graphed here) obeys the equation $$L \frac{d i}{d t}+R i=0,$$ which is Equation ( 5\()\) with \(V=0\) a. Solve the equation to express \(i\) as a function of \(t .\) b. How long after the switch is thrown will it take the current to fall to half its original value? c. Show that the value of the current when \(t=L / R\) is \(I / e\) . The significance of this time is explained in the next exercise.)

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0)\) (as in Example 5 ). Which equilibria are stable, and which are unstable? $$ \frac{d P}{d t}=P(1-2 P) $$

In Exercises 31 and \(32,\) obtain a slope field and graph the particular solution over the specified interval. Use your CAS DE solver to find the general solution of the differential equation. $$ \begin{array}{l}{\text { A logistic equation } y^{\prime}=y(2-y), \quad y(0)=1 / 2} \\ {\quad 0 \leq x \leq 4, \quad 0 \leq y \leq 3}\end{array} $$

An executive conference room of a corporation contains 4500 \(\mathrm{ft}^{3}\) of air initially free of carbon monoxide. Starting at time \(t=0,\) cigarette smoke containing 4\(\%\) carbon monoxide is blown into the room at the rate of \(0.3 \mathrm{ft}^{3} / \mathrm{min} .\) A ceiling fan keeps the air in the room well sirculated and the air leaves the room at the same rate of 0.3 \(\mathrm{ft}^{3} / \mathrm{min}\). Find the time when the concentration of carbon monoxide in the room reaches 0.01\(\%\).

Solve the differential equations. \(x y^{\prime}+3 y=\frac{\sin x}{x^{2}}, \quad x>0\)
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