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

The differential equation ( x 2 + y 2 ) d x + 2 x y d y = 0 . The solution of the differential equation ( x 2 + y 2 ) d x + 2 x y d y = 0 is x 3 + 3 x y 2 = C and x ≡ 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, ( x 2 + y 2 ) d x + 2 x y d y = 0 The equation can be rewritten as, 2 x y d y = − ( x 2 + y 2 ) d x d y d x = − x 2 + y 2 2 x y d y d x = − 1 2 ( x 2 + y 2 x y ) d y d x = − 1 2 ( x y + 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 = − 1 2 ( x y + y x ) v + x d v d x = − 1 2 ( 1 v + v ) x d v d x = − 1 2 ( 1 v + v ) − v x d v d x = ( − 1 2 v − v 2 − v ) Separate the variables and rewrite the differential equation as follows. x d v d x = − ( 2 + 2 v 2 + 4 v 2 4 v ) x d v d x = − ( 2 + 6 v 2 4 v ) x d v d x = − ( 1 + 3 v 2 ) 2 v 2 v 1 + 3 v 2 d v = − 1 x d x Integrate on both sides of the equation. ∫ 2 v 1 + 3 v 2 d v = ∫ ( − 1 x ) d x 1 3 ∫ 6 v 1 + 3 v 2 d v = − ln | x | + C 1 1 3 ln | 1 + 3 v 2 | = − ln | x | + C 1 1 3 ln | 1 + 3 v 2 | + ln | x | = C 1 Simplify further, ln | 1 + 3 v 2 | + 3 ln | x | = 3 C 1 ln | 1 + 3 v 2 | + ln | x 3 | = 3 C 1 ln | ( 1 + 3 v 2 ) ⋅ x 3 | = 3 C 1 ( 1 + 3 v 2 ) ⋅ x 3 = e 3 C 1 ( 1 + 3 v 2 ) ⋅ x 3 = C ( Take e 3 C 1 = C , C is an arbitrary constant ) Substitute back, v = y x for x ≠ 0 . ( 1 + 3 ( y x ) 2 ) ⋅ x 3 = C ( 1 + 3 y 2 x 2 ) ⋅ x 3 = C x 3 + 3 y 2 x 3 x 2 = C x 3 + 3 x y 2 = C Also, the equation may satisfied for x = 0 . So x ≡ 0 is another solution. Thus, the solutions of the differential equation ( x 2 + y 2 ) d x + 2 x y d y = 0 is x 3 + 3 x y 2 = C and x ≡ 0 _ .

The differential equation ( x 2 + y 2 ) d x + 2 x y d y = 0 . The solution of the differential equation ( x 2 + y 2 ) d x + 2 x y d y = 0 is x 3 + 3 x y 2 = C and x ≡ 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, ( x 2 + y 2 ) d x + 2 x y d y = 0 The equation can be rewritten as, 2 x y d y = − ( x 2 + y 2 ) d x d y d x = − x 2 + y 2 2 x y d y d x = − 1 2 ( x 2 + y 2 x y ) d y d x = − 1 2 ( x y + 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 = − 1 2 ( x y + y x ) v + x d v d x = − 1 2 ( 1 v + v ) x d v d x = − 1 2 ( 1 v + v ) − v x d v d x = ( − 1 2 v − v 2 − v ) Separate the variables and rewrite the differential equation as follows. x d v d x = − ( 2 + 2 v 2 + 4 v 2 4 v ) x d v d x = − ( 2 + 6 v 2 4 v ) x d v d x = − ( 1 + 3 v 2 ) 2 v 2 v 1 + 3 v 2 d v = − 1 x d x Integrate on both sides of the equation. ∫ 2 v 1 + 3 v 2 d v = ∫ ( − 1 x ) d x 1 3 ∫ 6 v 1 + 3 v 2 d v = − ln | x | + C 1 1 3 ln | 1 + 3 v 2 | = − ln | x | + C 1 1 3 ln | 1 + 3 v 2 | + ln | x | = C 1 Simplify further, ln | 1 + 3 v 2 | + 3 ln | x | = 3 C 1 ln | 1 + 3 v 2 | + ln | x 3 | = 3 C 1 ln | ( 1 + 3 v 2 ) ⋅ x 3 | = 3 C 1 ( 1 + 3 v 2 ) ⋅ x 3 = e 3 C 1 ( 1 + 3 v 2 ) ⋅ x 3 = C ( Take e 3 C 1 = C , C is an arbitrary constant ) Substitute back, v = y x for x ≠ 0 . ( 1 + 3 ( y x ) 2 ) ⋅ x 3 = C ( 1 + 3 y 2 x 2 ) ⋅ x 3 = C x 3 + 3 y 2 x 3 x 2 = C x 3 + 3 x y 2 = C Also, the equation may satisfied for x = 0 . So x ≡ 0 is another solution. Thus, the solutions of the differential equation ( x 2 + y 2 ) d x + 2 x y d y = 0 is x 3 + 3 x y 2 = C and x ≡ 0 _ .

Solution Summary: The author explains how to solve the differential equation (x2+y 2)dx+2xydy=0.

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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