Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problem 9.PE.11

Chapter 9, Problem 9.PE.11

In Exercises 1–22, solve the differential equation.

xy' + 2y = 1 - x⁻¹

Verified step by step guidance

Rewrite the given differential equation in the standard linear form. The equation is \(xy' + 2y = 1 - x^{-1}\). Divide both sides by \(x\) (assuming \(x \neq 0\)) to get \(y' + \frac{2}{x}y = \frac{1}{x} - \frac{1}{x^2}\).

Identify the integrating factor (IF) for the linear differential equation \(y' + P(x)y = Q(x)\), where \(P(x) = \frac{2}{x}\). The integrating factor is given by \(\mu(x) = e^{\int P(x)\,dx} = e^{\int \frac{2}{x} dx}\).

Calculate the integrating factor \(\mu(x) = e^{2 \ln|x|} = |x|^2\). Since \(x^2\) is positive for all \(x \neq 0\), we can write \(\mu(x) = x^2\).

Multiply the entire differential equation by the integrating factor \(x^2\) to obtain \(x^2 y' + 2x y = x^2 \left( \frac{1}{x} - \frac{1}{x^2} \right)\). Simplify the right side accordingly.

Recognize that the left side is the derivative of the product \(x^2 y\), so write \(\frac{d}{dx}(x^2 y) = x - 1\). Integrate both sides with respect to \(x\) to find \(x^2 y = \int (x - 1) dx + C\), where \(C\) is the constant of integration.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Linear Differential Equations

A first-order linear differential equation has the form y' + P(x)y = Q(x). Solving it involves finding an integrating factor to simplify the equation into an exact derivative, allowing integration to find the general solution.

Integrating Factor Method

The integrating factor is a function, usually denoted μ(x), used to multiply both sides of a linear differential equation to make the left side an exact derivative. It is typically μ(x) = e^(∫P(x) dx), facilitating straightforward integration.

Manipulating and Simplifying Differential Equations

Rearranging the given equation into standard form y' + P(x)y = Q(x) is essential. This often involves dividing through by coefficients and simplifying terms, such as handling expressions like x⁻¹, to correctly identify P(x) and Q(x) for solution.

Recommended video:

07:39

Classifying Differential Equations

Related Practice

Textbook Question

28. Derivation of Equation (7) in Example 4

a. Show that the solution of the equation

di /dt + R/Li = V/L

i = V/R + Cexp(-(R/L)i) .

Textbook Question

In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.

Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.

c. When will the tank have exactly 5 pounds of salt and how many gallons of solution will be in the tank?

views

Textbook Question

Solve the following initial value problem for u as a function of t:

du/dt + (k/m) u = 0 (k and m positive constants), u(0) = u₀

b. as a separable equation.

Textbook Question

In Exercises 1–22, solve the differential equation.

(x + 3y²) dy + y dx = 0 (Hint: d(xy) = y dx + x dy)

Textbook Question

In Exercises 1–22, solve the differential equation.

y' = sin³ x cos² y