Explanation Formula used: Integrating factor: The integrating factor of the First-Order linear differential equation y ′ + f ( x ) y = g ( x ) is I ( x ) = e ∫ f ( x ) d x . Theorem used: The solution of the differential equation, f ( y ) y ′ = g ( x ) is given implicitly by the equation, ∫ f ( y ) d y = ∫ g ( x ) d x . Calculation: Obtain the solution of the differential equation by using integrating factor. Rewrite the differential equation y ′ = 4 x + 2 x y 4 + x 2 as, y ′ = 4 x + 2 x y 4 + x 2 y ′ = 4 x 4 + x 2 + 2 x y 4 + x 2 y ′ − 2 x y 4 + x 2 = 4 x 4 + x 2 Comparing the equations, y ′ − 2 x y 4 + x 2 = 4 x 4 + x 2 and y ′ + f ( x ) y = g ( x ) , it is observed that f ( x ) = − 2 x 4 + x 2 and g ( x ) = 4 x 4 + x 2 . I ( x ) = e − ∫ 2 x 4 + x 2 d x = e − ln ( 4 + x 2 ) = e ln 1 ( 4 + x 2 ) = 1 4 + x 2 Thus, the integrating factor of the differential equation y ′ − 2 x y 4 + x 2 = 4 x 4 + x 2 is 1 4 + x 2 . Multiply both sides of the equation y ′ − 2 x y 4 + x 2 = 4 x 4 + x 2 by the integrating factor 1 4 + x 2 . ( 1 4 + x 2 ) y ′ − ( 1 4 + x 2 ) 2 x y 4 + x 2 = ( 1 4 + x 2 ) 4 x 4 + x 2 ( 1 4 + x 2 y ) ′ = 4 x ( 4 + x 2 ) 2 [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrate the above equation on both sides

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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 9.3, Problem 52E

To determine

To find: The general solution of the equation y′=4x+2xy4+x2 using integrating factor and separation of variables.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.3, Problem 51E

Chapter 9.3, Problem 53E

Students have asked these similar questions

Use mathematical induction to prove the following statement: For all natural numbers n, 5 divides 6^n - 1 (show every step in detail)

Solve this

Use mathematical induction to prove the following statement: For all natural numbers n, 5 divides 6^n - 1

Chapter 9 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

Ch. 9.1 - Show that y=Cx+1 is the general solution of the...Ch. 9.1 - Prob. 2MP Ch. 9.1 - Prob. 3MP Ch. 9.1 - Prob. 4MP Ch. 9.1 - Prob. 5MP Ch. 9.1 - Prob. 1ED Ch. 9.1 - Prob. 2ED Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E

Knowledge Booster

Learn more about

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

The general solution of the equation y ′ = 4 x + 2 x y 4 + x 2 using integrating factor and separation of variables.

Solution Summary: The author explains how to obtain the solution of the differential equation by using integrating factor and separation of variables.

Concept explainers

Videos

To find: The general solution of the equation y′=4x+2xy4+x2 using integrating factor and separation of variables.

Want to see the full answer?

Chapter 9 Solutions