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

The initial value problem d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 . The solution of the IVP d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 is y = ( 19 x 4 − 1 ) 2 _ . Definition used: “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, called a Bernoulli equation. For the values of n except 0 and 1, the substitution v = y 1 − n transforms the Bernoulli into a linear equation.” Calculation: The differential equation is, d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 It is observed that the given equation is a Bernoulli equation with n = − 1 , P ( x ) = − 2 x and Q ( x ) = x − 1 . First multiply the equation by y , to transform the equation into a linear equation as follows. y d y d x − y 2 y x = y x − 1 y − 1 y d y d x − 2 y 2 x = x − 1 y d y d x − 2 y 2 x = 1 x Next make the substitution v = y 2 and differentiate it implicitly. Since d v d x = 2 y d y d x , the transformed equation is, 1 2 d v d x − 2 x v = 1 x d v d x − 4 x v = 2 x ... ( 1 ) This is a linear equation of the standard form d y d x + P ( x ) y = Q ( x ) . Here, P ( x ) = − 4 x and Q ( x ) = 2 x . Calculate the integrating factor μ ( x ) by the formula μ ( x ) = exp [ ∫ P ( x ) d x ] . μ ( x ) = exp [ ∫ ( − 4 x ) d x ] = exp [ − 4 ln x ] = e ln x − 4 = x − 4 Multiply the equation (1) by μ ( x ) and obtain d d x ( μ ( x ) y ) = μ ( x ) Q ( x ) . x − 4 ( d v d x − 4 x v ) = x − 4 ( 2 x ) 1 x 4 d v d x − 1 x 4 ⋅ 4 x v = 1 x 4 ( 2 x ) 1 x 4 d v d x − 4 x 5 v = 2 x 5 d d x ( v x 4 ) = 2 x 5 To obtain the general solution y ( x ) = 1 μ ( x ) [ ∫ μ ( x ) Q ( x ) d x + C ] , integrate both sides of the equation d d x ( v x 4 ) = 2 x 5 and solve for y ( x ) by dividing μ ( x ) . v x 4 = ∫ ( 2 x 5 ) d x v x 4 = ∫ 2 x − 5 d x v x 4 = 2 x − 4 − 4 + C v = − 1 2 x − 4 x 4 + C x 4 v = − 1 2 + C x 4 Substitute back, v = y 2 gives the solution, y 2 = − 1 2 + C x 4 y = ± − 1 2 + C x 4 y = − 1 2 + C x 4 ( ∵ y ( 1 ) = 3 > 0 , neglect y = − − 1 2 + C x 4 ) Apply the initial condition x = 1 and y = 3 in the above result to find the constant C . ( 3 ) = − 1 2 + C ( 1 ) 4 ( 3 ) 2 = − 1 2 + C C = 9 + 1 2 C = 19 2 Substitute C = 19 2 in the solution y = − 1 2 + C x 4 and simplify further. y = − 1 2 + 19 2 x 4 y = ( 19 x 4 − 1 ) 2 Thus, the solution of the IVP d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 is y = ( 19 x 4 − 1 ) 2 _ .

The initial value problem d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 . The solution of the IVP d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 is y = ( 19 x 4 − 1 ) 2 _ . Definition used: “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, called a Bernoulli equation. For the values of n except 0 and 1, the substitution v = y 1 − n transforms the Bernoulli into a linear equation.” Calculation: The differential equation is, d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 It is observed that the given equation is a Bernoulli equation with n = − 1 , P ( x ) = − 2 x and Q ( x ) = x − 1 . First multiply the equation by y , to transform the equation into a linear equation as follows. y d y d x − y 2 y x = y x − 1 y − 1 y d y d x − 2 y 2 x = x − 1 y d y d x − 2 y 2 x = 1 x Next make the substitution v = y 2 and differentiate it implicitly. Since d v d x = 2 y d y d x , the transformed equation is, 1 2 d v d x − 2 x v = 1 x d v d x − 4 x v = 2 x ... ( 1 ) This is a linear equation of the standard form d y d x + P ( x ) y = Q ( x ) . Here, P ( x ) = − 4 x and Q ( x ) = 2 x . Calculate the integrating factor μ ( x ) by the formula μ ( x ) = exp [ ∫ P ( x ) d x ] . μ ( x ) = exp [ ∫ ( − 4 x ) d x ] = exp [ − 4 ln x ] = e ln x − 4 = x − 4 Multiply the equation (1) by μ ( x ) and obtain d d x ( μ ( x ) y ) = μ ( x ) Q ( x ) . x − 4 ( d v d x − 4 x v ) = x − 4 ( 2 x ) 1 x 4 d v d x − 1 x 4 ⋅ 4 x v = 1 x 4 ( 2 x ) 1 x 4 d v d x − 4 x 5 v = 2 x 5 d d x ( v x 4 ) = 2 x 5 To obtain the general solution y ( x ) = 1 μ ( x ) [ ∫ μ ( x ) Q ( x ) d x + C ] , integrate both sides of the equation d d x ( v x 4 ) = 2 x 5 and solve for y ( x ) by dividing μ ( x ) . v x 4 = ∫ ( 2 x 5 ) d x v x 4 = ∫ 2 x − 5 d x v x 4 = 2 x − 4 − 4 + C v = − 1 2 x − 4 x 4 + C x 4 v = − 1 2 + C x 4 Substitute back, v = y 2 gives the solution, y 2 = − 1 2 + C x 4 y = ± − 1 2 + C x 4 y = − 1 2 + C x 4 ( ∵ y ( 1 ) = 3 > 0 , neglect y = − − 1 2 + C x 4 ) Apply the initial condition x = 1 and y = 3 in the above result to find the constant C . ( 3 ) = − 1 2 + C ( 1 ) 4 ( 3 ) 2 = − 1 2 + C C = 9 + 1 2 C = 19 2 Substitute C = 19 2 in the solution y = − 1 2 + C x 4 and simplify further. y = − 1 2 + 19 2 x 4 y = ( 19 x 4 − 1 ) 2 Thus, the solution of the IVP d y d x − 2 y x = x − 1 y − 1 , y ( 1 ) = 3 is y = ( 19 x 4 − 1 ) 2 _ .

Solution Summary: The author explains that the given equation is a Bernoulli equation with n=-1,P(x)=-2x

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. 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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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