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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

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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

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