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

The general solution to the equation d r d θ = r 2 + 2 r θ θ 2 . The general solution of the differential equation d r d θ = r 2 + 2 r θ θ 2 is r = θ 2 C − θ and r ≡ 0 _ . 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 r d θ = r 2 + 2 r θ θ 2 d r d θ − 2 r θ θ 2 = r 2 θ 2 d r d θ − 2 r θ = r 2 θ 2 It is observed that the given equation is a Bernoulli equation with n = 2 , P ( θ ) = − 2 θ and Q ( θ ) = 1 θ 2 . First divide the equation by r 2 , to transform the equation into a linear equation as follows. 1 r 2 d r d θ − 2 r r 2 θ = r 2 r 2 θ 2 r − 2 d r d θ − 2 θ r − 1 = 1 θ 2 Next make the substitution v = r − 1 and differentiate it implicitly. Since d v d θ = − r − 2 d r d θ , the transformed equation is − d v d θ − 2 θ v = 1 θ 2 d v d θ + 2 θ v = − 1 θ 2 ... ( 1 ) This is a linear equation of the standard form d y d x + P ( x ) y = Q ( x ) . Here, P ( θ ) = 2 θ and Q ( θ ) = − 1 θ 2 . Calculate the integrating factor μ ( θ ) by the formula μ ( θ ) = exp [ ∫ P ( θ ) d θ ] . μ ( θ ) = exp [ ∫ ( 2 θ ) d θ ] = exp [ 2 ln θ ] = e ln θ 2 = θ 2 Multiply the equation (1) by μ ( θ ) and obtain d d θ ( μ ( θ ) v ) = μ ( θ ) Q ( θ ) . θ 2 ( d v d θ + 2 θ v ) = − θ 2 1 θ 2 θ 2 d v d θ + 2 θ v = − 1 d d θ ( θ 2 v ) = − 1 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 θ ( θ 2 v ) = − 1 and solve for v by dividing μ ( θ ) . θ 2 v = − 1 ∫ d θ + C θ 2 v = − θ + C v = C − θ θ 2 Substitute back, v = r − 1 in the above equation. r − 1 = C − θ θ 2 1 r = C − θ θ 2 r = θ 2 C − θ where r ≠ 0 ( ∵ the equation divided by r 2 in the first step ) Another solution is r ≡ 0 . Thus, the general solution of the differential equation d r d θ = r 2 + 2 r θ θ 2 is r = θ 2 C − θ and r ≡ 0 _ .

The general solution to the equation d r d θ = r 2 + 2 r θ θ 2 . The general solution of the differential equation d r d θ = r 2 + 2 r θ θ 2 is r = θ 2 C − θ and r ≡ 0 _ . 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 r d θ = r 2 + 2 r θ θ 2 d r d θ − 2 r θ θ 2 = r 2 θ 2 d r d θ − 2 r θ = r 2 θ 2 It is observed that the given equation is a Bernoulli equation with n = 2 , P ( θ ) = − 2 θ and Q ( θ ) = 1 θ 2 . First divide the equation by r 2 , to transform the equation into a linear equation as follows. 1 r 2 d r d θ − 2 r r 2 θ = r 2 r 2 θ 2 r − 2 d r d θ − 2 θ r − 1 = 1 θ 2 Next make the substitution v = r − 1 and differentiate it implicitly. Since d v d θ = − r − 2 d r d θ , the transformed equation is − d v d θ − 2 θ v = 1 θ 2 d v d θ + 2 θ v = − 1 θ 2 ... ( 1 ) This is a linear equation of the standard form d y d x + P ( x ) y = Q ( x ) . Here, P ( θ ) = 2 θ and Q ( θ ) = − 1 θ 2 . Calculate the integrating factor μ ( θ ) by the formula μ ( θ ) = exp [ ∫ P ( θ ) d θ ] . μ ( θ ) = exp [ ∫ ( 2 θ ) d θ ] = exp [ 2 ln θ ] = e ln θ 2 = θ 2 Multiply the equation (1) by μ ( θ ) and obtain d d θ ( μ ( θ ) v ) = μ ( θ ) Q ( θ ) . θ 2 ( d v d θ + 2 θ v ) = − θ 2 1 θ 2 θ 2 d v d θ + 2 θ v = − 1 d d θ ( θ 2 v ) = − 1 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 θ ( θ 2 v ) = − 1 and solve for v by dividing μ ( θ ) . θ 2 v = − 1 ∫ d θ + C θ 2 v = − θ + C v = C − θ θ 2 Substitute back, v = r − 1 in the above equation. r − 1 = C − θ θ 2 1 r = C − θ θ 2 r = θ 2 C − θ where r ≠ 0 ( ∵ the equation divided by r 2 in the first step ) Another solution is r ≡ 0 . Thus, the general solution of the differential equation d r d θ = r 2 + 2 r θ θ 2 is r = θ 2 C − θ and r ≡ 0 _ .

Solution Summary: The author explains the general solution of the differential equation dr dtheta, which is a Bernoulli equation.

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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