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Math EP FUND.OF DIFF.EQUATIONS-MYLAB (18 WK)

The integral curves for the system d y d t = − 4 x − y , d x d t = 2 x − y . The integral curves for the system d y d t = − 4 x − y , d x d t = 2 x − y is ( y − 4 x ) 2 ( y + x ) 3 = C _ . 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 given system of equation are, d y d t = − 4 x − y ... ( 1 ) d x d t = 2 x − y ... ( 2 ) Divide the equation (1) by (2).. d y d t d x d t = − 4 x − y 2 x − y d y d x = − 4 x − y 2 x − y Divide the right hand side of the numerator and the denominator by x . d y d x = − 4 x − y x 2 x − y x d y d x = − 4 − ( y x ) 2 − ( y x ) 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 = − 4 − v 2 − v x d v d x = − 4 − v 2 − v − v x d v d x = v 2 − 3 v − 4 2 − v Separate the variables and rewrite the differential equation as follows. 2 − v v 2 − 3 v − 4 d v = 1 x d x Integrate on both sides of the equation. ∫ 2 − v v 2 − 3 v − 4 d v = ∫ 1 x d x Consider the term 2 − v v 2 − 3 v − 4 . Split the term by using the method of partial fractions. 2 − v ( v − 4 ) ( v + 1 ) = A ( v − 4 ) + B ( v + 1 ) ... ( 3 ) Multiply ( v + 1 ) ( v − 4 ) on both sides of the equation. 2 − v = A ( v + 1 ) + B ( v − 4 ) Put v = 4 results A = − 2 5 , v = − 1 results B = − 3 5 . Substitute A = − 2 5 , and B = − 3 5 in the equation (3). 2 − v ( v + 1 ) ( v − 4 ) = − 2 5 ( v − 4 ) − 3 5 ( v + 1 ) Thus, the integral becomes ∫ 2 − v v 2 − 3 v − 4 d v = ∫ 1 x d x − 1 5 ∫ 2 ( v − 4 ) d v − 1 5 ∫ 3 ( v + 1 ) d v = ln x + C − 1 5 ( 2 ln | v − 4 | + 3 ln | v + 1 | ) = ln x + C ln | ( v − 4 ) 2 | + ln | ( v + 1 ) 3 | = − 5 ln x − 5 C 1 Simplify further, ln | ( v − 4 ) 2 ( v + 1 ) 3 | = ln x − 5 − 5 C 1 | ( v − 4 ) 2 ( v + 1 ) 3 | = e ln x − 5 − 5 C 1 ( v − 4 ) 2 ( v + 1 ) 3 = C x 5 Substitute back, v = y x for x ≠ 0 . ( y x − 4 ) 2 ( y x + 1 ) 3 = C x 5 ( y − 4 x x ) 2 ( y + x x ) 3 = C x 5 ( y − 4 x ) 2 ( y + x ) 3 x 2 x 3 = C x 5 ( y − 4 x ) 2 ( y + x ) 3 = C Thus, the integral curves for the system d y d t = − 4 x − y , d x d t = 2 x − y is ( y − 4 x ) 2 ( y + x ) 3 = C _ .

The integral curves for the system d y d t = − 4 x − y , d x d t = 2 x − y . The integral curves for the system d y d t = − 4 x − y , d x d t = 2 x − y is ( y − 4 x ) 2 ( y + x ) 3 = C _ . 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 given system of equation are, d y d t = − 4 x − y ... ( 1 ) d x d t = 2 x − y ... ( 2 ) Divide the equation (1) by (2).. d y d t d x d t = − 4 x − y 2 x − y d y d x = − 4 x − y 2 x − y Divide the right hand side of the numerator and the denominator by x . d y d x = − 4 x − y x 2 x − y x d y d x = − 4 − ( y x ) 2 − ( y x ) 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 = − 4 − v 2 − v x d v d x = − 4 − v 2 − v − v x d v d x = v 2 − 3 v − 4 2 − v Separate the variables and rewrite the differential equation as follows. 2 − v v 2 − 3 v − 4 d v = 1 x d x Integrate on both sides of the equation. ∫ 2 − v v 2 − 3 v − 4 d v = ∫ 1 x d x Consider the term 2 − v v 2 − 3 v − 4 . Split the term by using the method of partial fractions. 2 − v ( v − 4 ) ( v + 1 ) = A ( v − 4 ) + B ( v + 1 ) ... ( 3 ) Multiply ( v + 1 ) ( v − 4 ) on both sides of the equation. 2 − v = A ( v + 1 ) + B ( v − 4 ) Put v = 4 results A = − 2 5 , v = − 1 results B = − 3 5 . Substitute A = − 2 5 , and B = − 3 5 in the equation (3). 2 − v ( v + 1 ) ( v − 4 ) = − 2 5 ( v − 4 ) − 3 5 ( v + 1 ) Thus, the integral becomes ∫ 2 − v v 2 − 3 v − 4 d v = ∫ 1 x d x − 1 5 ∫ 2 ( v − 4 ) d v − 1 5 ∫ 3 ( v + 1 ) d v = ln x + C − 1 5 ( 2 ln | v − 4 | + 3 ln | v + 1 | ) = ln x + C ln | ( v − 4 ) 2 | + ln | ( v + 1 ) 3 | = − 5 ln x − 5 C 1 Simplify further, ln | ( v − 4 ) 2 ( v + 1 ) 3 | = ln x − 5 − 5 C 1 | ( v − 4 ) 2 ( v + 1 ) 3 | = e ln x − 5 − 5 C 1 ( v − 4 ) 2 ( v + 1 ) 3 = C x 5 Substitute back, v = y x for x ≠ 0 . ( y x − 4 ) 2 ( y x + 1 ) 3 = C x 5 ( y − 4 x x ) 2 ( y + x x ) 3 = C x 5 ( y − 4 x ) 2 ( y + x ) 3 x 2 x 3 = C x 5 ( y − 4 x ) 2 ( y + x ) 3 = C Thus, the integral curves for the system d y d t = − 4 x − y , d x d t = 2 x − y is ( y − 4 x ) 2 ( y + x ) 3 = C _ .

Solution Summary: The author explains how the integral curves for the system dyt=-4x-y,

EP FUND.OF DIFF.EQUATIONS-MYLAB (18 WK)

EP FUND.OF DIFF.EQUATIONS-MYLAB (18 WK)

9th Edition

ISBN: 9780135963777

Author: Nagle

Publisher: PEARSON CO

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EP FUND.OF DIFF.EQUATIONS-MYLAB (18 WK)

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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