Explanation Integrating factor: I ( x ) = e ∫ P ( x ) d x (1) Product rule: d ( f ⋅ g ) d x = f ⋅ d ( g ) d x + g ⋅ d ( f ) d x Calculation: Obtain the solution of the differential equation. Express the given differential equation as follows, Divide the differential equation by x 2 , x 2 y ′ − y x 2 = 2 x 3 e − 1 x x 2 x 2 y ′ x 2 − 1 x 2 y = 2 x 3 e − 1 x x 2 y ′ − 1 x 2 y = 2 x e − 1 x Thus, the redefined differential equation is y ′ − 1 x 2 y = 2 x e − 1 x . This is a first order linear differential equation of the form d y d x + P ( x ) y = Q ( x ) . Comparing the coefficients of y and constant terms, P ( x ) = − 1 x 2 and Q ( x ) = 2 x e − 1 x The integrating factor is computed as follows, Substitute P ( x ) = − 1 x 2 in equation (1), I ( x ) = e ∫ − 1 x 2 d x = e − ∫ x − 2 d x = e − ( x − 2 + 1 − 2 + 1 ) </

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 9, Problem 8RE

To determine

To solve: The differential equation x2y′−y=2x3e−1x .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9, Problem 7RE

Chapter 9, Problem 9RE

Students have asked these similar questions

The spread of an infectious disease is often modeled using the following autonomous differential equation: dI - - BI(N − I) − MI, dt where I is the number of infected people, N is the total size of the population being modeled, ẞ is a constant determining the rate of transmission, and μ is the rate at which people recover from infection. Close a) (5 points) Suppose ẞ = 0.01, N = 1000, and µ = 2. Find all equilibria. b) (5 points) For the equilbria in part a), determine whether each is stable or unstable. c) (3 points) Suppose ƒ(I) = d. Draw a phase plot of f against I. (You can use Wolfram Alpha or Desmos to plot the function, or draw the dt function by hand.) Identify the equilibria as stable or unstable in the graph. d) (2 points) Explain the biological meaning of these equilibria being stable or unstable.

Find the indefinite integral. Check Answer: 7x 4 + 1x dx

show sketch

Chapter 9 Solutions

Single Variable Calculus

Ch. 9.1 - Show that y=23ex+e2x is a solution of the...Ch. 9.1 - Verify that y = t cos t t is a solution of the...Ch. 9.1 - (a) For what values of r does the function y = erx...Ch. 9.1 - (a) For what values of k does the function y = cos...Ch. 9.1 - Which of the following functions are solutions of...Ch. 9.1 - Prob. 6E Ch. 9.1 - (a) What can you say about a solution of the...Ch. 9.1 - (a) What can you say about the graph of a solution...Ch. 9.1 - A population is modeled by the differential...Ch. 9.1 - The Fitzhugh-Nagumo model for the electrical...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The differential equation x 2 y ′ − y = 2 x 3 e − 1 x .

Solution Summary: The author explains how to solve the differential equation by dividing it by x2 and obtaining the solution.

Concept explainers

Videos

To solve: The differential equation x2y′−y=2x3e−1x .

Want to see the full answer?

Chapter 9 Solutions