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

Whether the differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 as homogeneous, Bernoulli, linear coefficients, or of the form y ′ = G ( a x + b y ) . The differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 can be identified as Homogeneous and Bernoulli _ . Definition used: Homogeneous equation: “If the right-hand side of the equation d y d x = f ( x , y ) can be expressed as a function of the ratio y x alone, then we say the equation is homogeneous.” Bernoulli equation: “A first-order equation that can be written in the form d y d x + P ( x ) y = Q ( x ) y n , where P ( x ) and Q ( x ) are continuous on an interval ( a , b ) and n is a real number, is called a Bernoulli equation.” Equations of the form d y d x = G ( a x + b y ) : “When the right hand side of the equation d y d x = f ( x , y ) can be expressed as a function of the combination a x + b y , where a and b are constants, that is, d y d x = G ( a x + b y ) .” Linear coefficients: “The equations of the form ( a 1 x + b 1 y + c 1 ) d x + ( a 2 x + b 2 y + c 2 d y ) = 0 , where a i 's, b i 's, and c i 's are constants. If a 1 b 2 ≠ a 2 b 1 and b 1 ≠ a 2 , seek a translation of axes of the form x = u + h and y = v + k , where h and k satisfy a 1 h + b 1 k + c 1 = 0 , and a 2 h + b 2 k + c 2 = 0 .” Calculation: The differential equation is, ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 Rewrite the equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 as shown below. 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 ... ( 1 ) d y d θ = 1 2 ( − ( y θ ) 2 + y θ ) Note that, the right hand side of the equation is of the form d y d θ = f ( y , θ ) and it can be expressed as a function of the ratio y θ alone. Thus, the equation can be identified as homogenous. Rewrite the equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 as shown below. ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 d y d θ = − y 2 2 θ 2 + y 2 θ d y d θ − 1 2 θ y = − 1 2 θ 2 y 2 Note that, the equation is of the form d y d θ + P ( θ ) y = Q ( θ ) y n . This is a Bernoulli equation with n = 2 , P ( θ ) = − 1 2 θ and Q ( θ ) = − 1 2 θ 2 . Note that, the equation cannot be rewritten of the form d y d x = G ( a x + b y ) and it is not of the form ( a 1 x + b 1 y + c 1 ) d x + ( a 2 x + b 2 y + c 2 d y ) = 0 . By the definitions stated above, the equation cannot be identified as Linear coefficients and the equations of the form d y d x = G ( a x + b y ) . Therefore, the equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 can be identified as Homogeneous and Bernoulli _ .

Whether the differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 as homogeneous, Bernoulli, linear coefficients, or of the form y ′ = G ( a x + b y ) . The differential equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 can be identified as Homogeneous and Bernoulli _ . Definition used: Homogeneous equation: “If the right-hand side of the equation d y d x = f ( x , y ) can be expressed as a function of the ratio y x alone, then we say the equation is homogeneous.” Bernoulli equation: “A first-order equation that can be written in the form d y d x + P ( x ) y = Q ( x ) y n , where P ( x ) and Q ( x ) are continuous on an interval ( a , b ) and n is a real number, is called a Bernoulli equation.” Equations of the form d y d x = G ( a x + b y ) : “When the right hand side of the equation d y d x = f ( x , y ) can be expressed as a function of the combination a x + b y , where a and b are constants, that is, d y d x = G ( a x + b y ) .” Linear coefficients: “The equations of the form ( a 1 x + b 1 y + c 1 ) d x + ( a 2 x + b 2 y + c 2 d y ) = 0 , where a i 's, b i 's, and c i 's are constants. If a 1 b 2 ≠ a 2 b 1 and b 1 ≠ a 2 , seek a translation of axes of the form x = u + h and y = v + k , where h and k satisfy a 1 h + b 1 k + c 1 = 0 , and a 2 h + b 2 k + c 2 = 0 .” Calculation: The differential equation is, ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 Rewrite the equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 as shown below. 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 ... ( 1 ) d y d θ = 1 2 ( − ( y θ ) 2 + y θ ) Note that, the right hand side of the equation is of the form d y d θ = f ( y , θ ) and it can be expressed as a function of the ratio y θ alone. Thus, the equation can be identified as homogenous. Rewrite the equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 as shown below. ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 d y d θ = − y 2 2 θ 2 + y 2 θ d y d θ − 1 2 θ y = − 1 2 θ 2 y 2 Note that, the equation is of the form d y d θ + P ( θ ) y = Q ( θ ) y n . This is a Bernoulli equation with n = 2 , P ( θ ) = − 1 2 θ and Q ( θ ) = − 1 2 θ 2 . Note that, the equation cannot be rewritten of the form d y d x = G ( a x + b y ) and it is not of the form ( a 1 x + b 1 y + c 1 ) d x + ( a 2 x + b 2 y + c 2 d y ) = 0 . By the definitions stated above, the equation cannot be identified as Linear coefficients and the equations of the form d y d x = G ( a x + b y ) . Therefore, the equation ( y 3 − θ y 2 ) d θ + 2 θ 2 y d y = 0 can be identified as Homogeneous and Bernoulli _ .

Solution Summary: The author explains the difference between homogeneous and Bernoulli equations.

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. 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