Skip to main content

Math Advanced Math Fundamentals of Differential Equations (9th Edition)

The differential equation d y d x = 3 x y 2 x 2 − y 2 and sketch several of these lines. The solution of the differential equation d y d x = 3 x y 2 x 2 − y 2 is y 4 − C ( y 2 + x 2 ) 3 = 0 _ . Definition used: “The homogeneous equation is of the form d y d x = G ( y x ) . Let v = y x . Then d y d x = v + x ( d v d x ) , and the transformed equation in the variables v and x is separable.” Calculation: The differential equation is, d y d x = 3 x y 2 x 2 − y 2 The equation can be rewritten as, d y d x = 3 x y 2 x 2 − y 2 d y d x = 3 x y x 2 2 x 2 − y 2 x 2 d y d x = 3 ( y x ) 2 − ( y x ) 2 This is homogeneous equation of the form d y d x = G ( y x ) . So take v = y x . Then d y d x = v + x d v d x , the transformed equation is v + x d v d x = 3 v 2 − v 2 x d v d x = 3 v 2 − v 2 − v x d v d x = 3 v − 2 v + v 3 2 − v 2 x d v d x = v 3 + v 2 − v 2 Separate the variables and rewrite the differential equation as follows. 2 − v 2 v 3 + v d v = 1 x d x Integrate on both sides of the equation. ∫ 2 − v 2 v 3 + v d v = ∫ 1 x d x ∫ 2 − v 2 v ( v 2 + 1 ) d v = ∫ 1 x d x Consider the term 2 − v 2 v ( v 2 + 1 ) . Split the term by using the method of partial fractions. 2 − v 2 v ( v 2 + 1 ) = A v + C v + B ( v 2 + 1 ) ... ( 1 ) Multiply ( v ) ( v 2 + 1 ) on both sides of the equation. 2 − v 2 = A ( v 2 + 1 ) + ( C v + B ) ( v ) Equate the coefficients of constants, v , and v 2 . { 2 = A 0 = B − 1 = A + C ⇒ C = − 3 Substitute A = 2 , B = 0 and C = − 3 in the equation (1). 2 − v 2 v ( v 2 + 1 ) = 2 v − 3 v ( v 2 + 1 ) Thus, the integral becomes ∫ ( 2 v − 3 v ( v 2 + 1 ) ) d v = ∫ 1 x d x 2 ∫ 1 v d v − 3 2 ∫ 2 v ( v 2 + 1 ) d v = ∫ 1 x d x 2 ln | v | − 3 2 ln | v 2 + 1 | = ln | x | + C 1 4 ln | v | − 3 ln | v 2 + 1 | = 2 ln | x | + 2 C 1 Simplify further, ln | v 4 | − ln | ( v 2 + 1 ) 3 | − ln | x 2 | = 2 C 1 ln | v 4 x 2 ( v 2 + 1 ) 3 | = 2 C 1 v 4 x 2 ( v 2 + 1 ) 3 = e 2 C 1 v 4 x 2 ( v 2 + 1 ) 3 = C ( C is an arbitrary constant ) Substitute back, v = y x for x ≠ 0 . ( y x ) 4 x 2 ( ( y x ) 2 + 1 ) 3 = C y 4 x 4 x 2 ( y 2 + x 2 x 2 ) 3 = C x 6 y 4 x 6 ( y 2 + x 2 ) 3 = C y 4 ( y 2 + x 2 ) 3 = C Simplify further, y 4 = C ( y 2 + x 2 ) 3 y 4 − C ( y 2 + x 2 ) 3 = 0 Thus, the solution of the differential equation d y d x = 3 x y 2 x 2 − y 2 is y 4 − C ( y 2 + x 2 ) 3 = 0 _ . Use online graphing calculator to sketch the solution y 4 − C ( y 2 + x 2 ) 3 = 0 for the several values of C as shown in Figure 1. From figure 1, it is verified that the magnetic field lines of a dipole satisfy d y d x = 3 x y 2 x 2 − y 2 .

The differential equation d y d x = 3 x y 2 x 2 − y 2 and sketch several of these lines. The solution of the differential equation d y d x = 3 x y 2 x 2 − y 2 is y 4 − C ( y 2 + x 2 ) 3 = 0 _ . Definition used: “The homogeneous equation is of the form d y d x = G ( y x ) . Let v = y x . Then d y d x = v + x ( d v d x ) , and the transformed equation in the variables v and x is separable.” Calculation: The differential equation is, d y d x = 3 x y 2 x 2 − y 2 The equation can be rewritten as, d y d x = 3 x y 2 x 2 − y 2 d y d x = 3 x y x 2 2 x 2 − y 2 x 2 d y d x = 3 ( y x ) 2 − ( y x ) 2 This is homogeneous equation of the form d y d x = G ( y x ) . So take v = y x . Then d y d x = v + x d v d x , the transformed equation is v + x d v d x = 3 v 2 − v 2 x d v d x = 3 v 2 − v 2 − v x d v d x = 3 v − 2 v + v 3 2 − v 2 x d v d x = v 3 + v 2 − v 2 Separate the variables and rewrite the differential equation as follows. 2 − v 2 v 3 + v d v = 1 x d x Integrate on both sides of the equation. ∫ 2 − v 2 v 3 + v d v = ∫ 1 x d x ∫ 2 − v 2 v ( v 2 + 1 ) d v = ∫ 1 x d x Consider the term 2 − v 2 v ( v 2 + 1 ) . Split the term by using the method of partial fractions. 2 − v 2 v ( v 2 + 1 ) = A v + C v + B ( v 2 + 1 ) ... ( 1 ) Multiply ( v ) ( v 2 + 1 ) on both sides of the equation. 2 − v 2 = A ( v 2 + 1 ) + ( C v + B ) ( v ) Equate the coefficients of constants, v , and v 2 . { 2 = A 0 = B − 1 = A + C ⇒ C = − 3 Substitute A = 2 , B = 0 and C = − 3 in the equation (1). 2 − v 2 v ( v 2 + 1 ) = 2 v − 3 v ( v 2 + 1 ) Thus, the integral becomes ∫ ( 2 v − 3 v ( v 2 + 1 ) ) d v = ∫ 1 x d x 2 ∫ 1 v d v − 3 2 ∫ 2 v ( v 2 + 1 ) d v = ∫ 1 x d x 2 ln | v | − 3 2 ln | v 2 + 1 | = ln | x | + C 1 4 ln | v | − 3 ln | v 2 + 1 | = 2 ln | x | + 2 C 1 Simplify further, ln | v 4 | − ln | ( v 2 + 1 ) 3 | − ln | x 2 | = 2 C 1 ln | v 4 x 2 ( v 2 + 1 ) 3 | = 2 C 1 v 4 x 2 ( v 2 + 1 ) 3 = e 2 C 1 v 4 x 2 ( v 2 + 1 ) 3 = C ( C is an arbitrary constant ) Substitute back, v = y x for x ≠ 0 . ( y x ) 4 x 2 ( ( y x ) 2 + 1 ) 3 = C y 4 x 4 x 2 ( y 2 + x 2 x 2 ) 3 = C x 6 y 4 x 6 ( y 2 + x 2 ) 3 = C y 4 ( y 2 + x 2 ) 3 = C Simplify further, y 4 = C ( y 2 + x 2 ) 3 y 4 − C ( y 2 + x 2 ) 3 = 0 Thus, the solution of the differential equation d y d x = 3 x y 2 x 2 − y 2 is y 4 − C ( y 2 + x 2 ) 3 = 0 _ . Use online graphing calculator to sketch the solution y 4 − C ( y 2 + x 2 ) 3 = 0 for the several values of C as shown in Figure 1. From figure 1, it is verified that the magnetic field lines of a dipole satisfy d y d x = 3 x y 2 x 2 − y 2 .

Solution Summary: The author explains how to solve the differential equation dy dx, which is a homogeneous equation.

Fundamentals of Differential Equations (9th Edition)

Fundamentals of Differential Equations (9th Edition)

9th Edition

ISBN: 9780321977069

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

Blurred answer

Chapter 2.6, Problem 45E

Chapter 2.6, Problem 47E

Students have asked these similar questions

Numerical an

7 Through the method of inverting the coefficient matrix, solve x+2x=8 2x - y = -5 x+y+z = 4.

Numerical ana

Chapter 2 Solutions

Fundamentals of Differential Equations (9th Edition)

Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 716, solve the equation. 7. xdydx=1y3 Ch. 2.2 - In Problems 716, solve the equation. 8. dxdt=3xt2 Ch. 2.2 - In Problems 716, solve the equation. 9....Ch. 2.2 - In Problems 716, solve the equation. 10....

Ch. 2.2 - In Problems 716, solve the equation. 11....Ch. 2.2 - In Problems 716, solve the equation. 12....Ch. 2.2 - In Problems 716, solve the equation. 13....Ch. 2.2 - In Problems 716, solve the equation. 14. dxdtx3=x Ch. 2.2 - In Problems 716, solve the equation. 15....Ch. 2.2 - In Problems 716, solve the equation. 16. y1 dy +...Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - Prob. 23E Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - Prob. 27E Ch. 2.2 - Sketch the solution to the initial value problem...Ch. 2.2 - Uniqueness Questions. In Chapter 1 we indicated...Ch. 2.2 - As stated in this section, the separation of...Ch. 2.2 - Prob. 31E Ch. 2.2 - Prob. 32E Ch. 2.2 - Mixing. Suppose a brine containing 0.3 kilogram...Ch. 2.2 - Newtons Law of Cooling. According to Newtons law...Ch. 2.2 - Prob. 35E Ch. 2.2 - Prob. 36E Ch. 2.2 - Compound Interest. If P(t) is the amount of...Ch. 2.2 - Free Fall. In Section 2.1, we discussed a model...Ch. 2.2 - Grand Prix Race. Driver A had been leading...Ch. 2.2 - Prob. 40E Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - Prob. 15E Ch. 2.3 - Prob. 17E Ch. 2.3 - Prob. 18E Ch. 2.3 - Prob. 19E Ch. 2.3 - Prob. 20E Ch. 2.3 - In Problems 1722, solve the initial value problem....Ch. 2.3 - In Problems 1722, solve the initial value problem....Ch. 2.3 - Radioactive Decay. In Example 2 assume that the...Ch. 2.3 - Prob. 24E Ch. 2.3 - (a) Using definite integration, show that the...Ch. 2.3 - Prob. 26E Ch. 2.3 - Constant Multiples of Solutions. (a) Show that y =...Ch. 2.3 - Prob. 29E Ch. 2.3 - Bernoulli Equations. The equation (18) dydx+2y=xy2...Ch. 2.3 - Prob. 31E Ch. 2.3 - Prob. 32E Ch. 2.3 - Prob. 33E Ch. 2.3 - Prob. 34E Ch. 2.3 - Prob. 35E Ch. 2.3 - Prob. 36E Ch. 2.3 - Prob. 37E Ch. 2.3 - Prob. 38E Ch. 2.3 - Prob. 39E Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - Prob. 3E Ch. 2.4 - Prob. 4E Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - Prob. 9E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 11E Ch. 2.4 - Prob. 12E Ch. 2.4 - Prob. 13E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 15E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 17E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 19E Ch. 2.4 - Prob. 20E Ch. 2.4 - In Problems 2126, solve the initial value problem....Ch. 2.4 - Prob. 22E Ch. 2.4 - Prob. 23E Ch. 2.4 - In Problems 2126, solve the initial value problem....Ch. 2.4 - Prob. 25E Ch. 2.4 - In Problems 2126, solve the initial value problem....Ch. 2.4 - Prob. 27E Ch. 2.4 - For each of the following equations, find the most...Ch. 2.4 - Prob. 29E Ch. 2.4 - Prob. 30E Ch. 2.4 - Prob. 31E Ch. 2.4 - Orthogonal Trajectories. A geometric problem...Ch. 2.4 - Prob. 33E Ch. 2.4 - Prob. 34E Ch. 2.4 - Prob. 35E Ch. 2.4 - Prob. 36E Ch. 2.5 - Prob. 1E Ch. 2.5 - In Problems 16, identify the equation as...Ch. 2.5 - Prob. 3E Ch. 2.5 - Prob. 4E Ch. 2.5 - In Problems 16, identify the equation as...Ch. 2.5 - Prob. 6E Ch. 2.5 - Prob. 7E Ch. 2.5 - Prob. 8E Ch. 2.5 - Prob. 9E Ch. 2.5 - Prob. 10E Ch. 2.5 - Prob. 11E Ch. 2.5 - Prob. 12E Ch. 2.5 - Prob. 13E Ch. 2.5 - Prob. 14E Ch. 2.5 - Prob. 15E Ch. 2.5 - Prob. 16E Ch. 2.5 - Prob. 17E Ch. 2.5 - Prob. 18E Ch. 2.5 - Prob. 19E Ch. 2.5 - Verify that when the linear differential equation...Ch. 2.6 - In Problems 18, identify (do not solve) the...Ch. 2.6 - Prob. 2E Ch. 2.6 - Prob. 3E Ch. 2.6 - Prob. 4E Ch. 2.6 - Prob. 5E Ch. 2.6 - Prob. 6E Ch. 2.6 - In Problems 18, identify (do not solve) the...Ch. 2.6 - Prob. 8E Ch. 2.6 - Use the method discussed under Homogeneous...Ch. 2.6 - Prob. 10E Ch. 2.6 - Prob. 11E Ch. 2.6 - Prob. 12E Ch. 2.6 - Prob. 13E Ch. 2.6 - Prob. 14E Ch. 2.6 - Prob. 15E Ch. 2.6 - Prob. 16E Ch. 2.6 - Prob. 17E Ch. 2.6 - Prob. 18E Ch. 2.6 - Prob. 19E Ch. 2.6 - Prob. 20E Ch. 2.6 - Prob. 21E Ch. 2.6 - Prob. 22E Ch. 2.6 - Use the method discussed under Bernoulli Equations...Ch. 2.6 - Prob. 24E Ch. 2.6 - Prob. 25E Ch. 2.6 - Prob. 26E Ch. 2.6 - Prob. 27E Ch. 2.6 - Prob. 28E Ch. 2.6 - Use the method discussed under Equations with...Ch. 2.6 - Prob. 30E Ch. 2.6 - Prob. 31E Ch. 2.6 - Prob. 32E Ch. 2.6 - Prob. 33E Ch. 2.6 - Prob. 34E Ch. 2.6 - Prob. 35E Ch. 2.6 - In Problems 3340, solve the equation given in: 36....Ch. 2.6 - Prob. 37E Ch. 2.6 - Prob. 38E Ch. 2.6 - Prob. 39E Ch. 2.6 - Prob. 40E Ch. 2.6 - Prob. 41E Ch. 2.6 - Prob. 42E Ch. 2.6 - Prob. 43E Ch. 2.6 - Show that equation (13) reduces to an equation of...Ch. 2.6 - Prob. 45E Ch. 2.6 - Prob. 46E Ch. 2.6 - Prob. 47E Ch. 2.6 - Prob. 48E Ch. 2 - In Problems 130, solve the equation. 1....Ch. 2 - Prob. 2RP Ch. 2 - Prob. 3RP Ch. 2 - Prob. 4RP Ch. 2 - Prob. 5RP Ch. 2 - In Problems 130, solve the equation. 6. 2xy3 dx ...Ch. 2 - In Problems 130, solve the equation. 7. t3y2 dt +...Ch. 2 - Prob. 8RP Ch. 2 - In Problems 130, solve the equation. 9. (x2 + y2)...Ch. 2 - Prob. 10RP Ch. 2 - Prob. 11RP Ch. 2 - Prob. 12RP Ch. 2 - Prob. 13RP Ch. 2 - Prob. 14RP Ch. 2 - Prob. 15RP Ch. 2 - Prob. 16RP Ch. 2 - Prob. 17RP Ch. 2 - Prob. 18RP Ch. 2 - Prob. 19RP Ch. 2 - Prob. 20RP Ch. 2 - Prob. 21RP Ch. 2 - Prob. 22RP Ch. 2 - Prob. 23RP Ch. 2 - In Problems 130, solve the equation. 24. (y/x +...Ch. 2 - Prob. 25RP Ch. 2 - Prob. 26RP Ch. 2 - Prob. 27RP Ch. 2 - Prob. 28RP Ch. 2 - Prob. 29RP Ch. 2 - Prob. 30RP Ch. 2 - Prob. 31RP Ch. 2 - Prob. 32RP Ch. 2 - Prob. 33RP Ch. 2 - Prob. 34RP Ch. 2 - Prob. 35RP Ch. 2 - Prob. 36RP Ch. 2 - Prob. 37RP Ch. 2 - Prob. 38RP Ch. 2 - Prob. 39RP Ch. 2 - Prob. 40RP Ch. 2 - Prob. 41RP

Knowledge Booster

Background pattern image

$Advanced Math$

Learn more about

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

Similar questions

SEE MORE QUESTIONS

Recommended textbooks for you

Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,
Basic Technical Mathematics
Advanced Math
ISBN:9780134437705
Author:Washington
Publisher:PEARSON
Topology
Advanced Math
ISBN:9780134689517
Author:Munkres, James R.
Publisher:Pearson,

Text book image

Advanced Engineering Mathematics

Advanced Math

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Wiley, John & Sons, Incorporated

Text book image

Numerical Methods for Engineers

Advanced Math

ISBN:9780073397924

Author:Steven C. Chapra Dr., Raymond P. Canale

Publisher:McGraw-Hill Education

Text book image

Introductory Mathematics for Engineering Applicat...

Advanced Math

ISBN:9781118141809

Author:Nathan Klingbeil

Publisher:WILEY

Text book image

Mathematics For Machine Technology

Advanced Math

ISBN:9781337798310

Author:Peterson, John.

Publisher:Cengage Learning,

Text book image

Basic Technical Mathematics

Advanced Math

ISBN:9780134437705

Author:Washington

Publisher:PEARSON

Text book image

Advanced Math

ISBN:9780134689517

Author:Munkres, James R.

Publisher:Pearson,

SEE MORE TEXTBOOKS

01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY

Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY

Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY