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

The given equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 in Problem 8. The solution of the differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 is y = C θ θ − C _ . Definition used: “The homogeneous equation is of the form d y d x = G ( y x ) . Let v = y θ . Then d y d θ = v + θ ( d v d θ ) , and the transformed equation in the variables v and x is separable.” Calculation: The differential equation is, ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 The equation can be rewritten as, 2 θ 2 y d y = − ( y 3 − θ y 2 ) d θ 2 θ 2 y d y d θ = − y 3 + θ y 2 d y d θ = − y 3 2 θ 2 y + θ y 2 2 θ 2 y d y d θ = − y 2 2 θ 2 + y 2 θ d y d θ = − 1 2 ( y θ ) 2 + 1 2 ( y θ ) This is homogeneous equation of the form d y d x = G ( y x ) . So, take v = y θ . Then d y d θ = v + θ d v d θ , the transformed equation is, v + θ d v d θ = − 1 2 ( v ) 2 + 1 2 v θ d v d θ = − 1 2 v 2 + 1 2 v − v θ d v d θ = − 1 2 v 2 − 1 2 v θ d v d θ = − 1 2 ( v 2 + v ) Separate the variables and rewrite the differential equation as follows. 1 v 2 + v d v = − 1 2 θ d θ 1 v ( v + 1 ) d v = − 1 2 θ d θ Integrate on both sides of the equation. ∫ 1 v ( v + 1 ) d v = − ∫ 1 2 θ d θ ... ( 1 ) Consider the term 1 v ( v + 1 ) . Split the term by using the method of partial fractions. 1 v ( v + 1 ) = A v + B v + 1 ... ( 2 ) Multiply ( v ) ( v + 1 ) on both sides of the equation. 1 = A ( v + 1 ) + B ( v ) Put v = 0 results A = 1 and put v = − 1 results B = − 1 . Substitute A = 1 and B = − 1 in the equation (2). 1 v ( v + 1 ) = 1 v − 1 v + 1 Thus, the integral (4) becomes, ∫ [ 1 v − 1 v + 1 ] d v = − ∫ 1 2 θ d θ ∫ 1 v d v − ∫ 1 v + 1 d v = − 1 2 ∫ 1 θ d θ ln | v | − ln | v + 1 | = − 1 2 ln | θ | + C 1 ln | v v + 1 | = ln | 1 θ | + C 1 Simplify further as follows. ln | v v + 1 | − ln | 1 θ | = C 1 ln | v θ v + 1 | = C 1 v θ v + 1 = e C 1 v θ v + 1 = C ( C is an arbitrary constant ) Substitute back, v = y θ . ( y θ ) θ ( y θ ) + 1 = C y θ y + θ = C y θ = C y + C θ y θ − C y = C θ y = C θ θ − C Thus, the solution of the differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 is y = C θ θ − C _ .

The given equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 in Problem 8. The solution of the differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 is y = C θ θ − C _ . Definition used: “The homogeneous equation is of the form d y d x = G ( y x ) . Let v = y θ . Then d y d θ = v + θ ( d v d θ ) , and the transformed equation in the variables v and x is separable.” Calculation: The differential equation is, ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 The equation can be rewritten as, 2 θ 2 y d y = − ( y 3 − θ y 2 ) d θ 2 θ 2 y d y d θ = − y 3 + θ y 2 d y d θ = − y 3 2 θ 2 y + θ y 2 2 θ 2 y d y d θ = − y 2 2 θ 2 + y 2 θ d y d θ = − 1 2 ( y θ ) 2 + 1 2 ( y θ ) This is homogeneous equation of the form d y d x = G ( y x ) . So, take v = y θ . Then d y d θ = v + θ d v d θ , the transformed equation is, v + θ d v d θ = − 1 2 ( v ) 2 + 1 2 v θ d v d θ = − 1 2 v 2 + 1 2 v − v θ d v d θ = − 1 2 v 2 − 1 2 v θ d v d θ = − 1 2 ( v 2 + v ) Separate the variables and rewrite the differential equation as follows. 1 v 2 + v d v = − 1 2 θ d θ 1 v ( v + 1 ) d v = − 1 2 θ d θ Integrate on both sides of the equation. ∫ 1 v ( v + 1 ) d v = − ∫ 1 2 θ d θ ... ( 1 ) Consider the term 1 v ( v + 1 ) . Split the term by using the method of partial fractions. 1 v ( v + 1 ) = A v + B v + 1 ... ( 2 ) Multiply ( v ) ( v + 1 ) on both sides of the equation. 1 = A ( v + 1 ) + B ( v ) Put v = 0 results A = 1 and put v = − 1 results B = − 1 . Substitute A = 1 and B = − 1 in the equation (2). 1 v ( v + 1 ) = 1 v − 1 v + 1 Thus, the integral (4) becomes, ∫ [ 1 v − 1 v + 1 ] d v = − ∫ 1 2 θ d θ ∫ 1 v d v − ∫ 1 v + 1 d v = − 1 2 ∫ 1 θ d θ ln | v | − ln | v + 1 | = − 1 2 ln | θ | + C 1 ln | v v + 1 | = ln | 1 θ | + C 1 Simplify further as follows. ln | v v + 1 | − ln | 1 θ | = C 1 ln | v θ v + 1 | = C 1 v θ v + 1 = e C 1 v θ v + 1 = C ( C is an arbitrary constant ) Substitute back, v = y θ . ( y θ ) θ ( y θ ) + 1 = C y θ y + θ = C y θ = C y + C θ y θ − C y = C θ y = C θ θ − C Thus, the solution of the differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 is y = C θ θ − C _ .

Solution Summary: The author explains the solution of the differential equation (y3-theta

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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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

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