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

(a) To show: That x y ′ + 2 y = 3 x has only one solution defined at x = 0 . Then show that the initial value problem for this equation with initial condition y ( 0 ) = y 0 has a unique solution when y 0 = 0 and no solution when y 0 ≠ 0 . Given: The initial value problem is, d y d x + P ( x ) y = x , y ( 0 ) = 1 where P ( x ) = { 1 , 0 ≤ x ≤ 2 , 3 , x > 2. Calculation: Consider the differential equation. x y ′ + 2 y = 3 x Assume that the solution is defined at the point x = 0 and does not defined at the domain. Rewrite the differential equation as follows. x y ′ x + 2 y x = 3 x x y ′ + 2 x y = 3 d y d x + 2 x y = 3 ... ( 1 ) Here, P ( x ) = 2 x and Q ( x ) = 3 . Calculate the integrating factor μ ( x ) by the formula μ ( x ) = exp [ ∫ P ( x ) d x ] . μ ( x ) = exp [ ∫ ( 2 x ) d x ] = e 2 ln x = e ln x 2 = x 2 Multiply the equation (1) by μ ( x ) and obtain d d x ( μ ( x ) y ) = μ ( x ) Q ( x ) . x 2 ( d y d x + 2 x y ) = 3 x 2 x 2 d y d x + 2 x 2 x y = 3 x 2 x 2 d y d x + 2 x y = 3 x 2 d d x ( x 2 y ) = 3 x 2 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 ( x 2 y ) = 3 x 2 and solve for y ( x ) by dividing μ ( x ) . x 2 y = ∫ 3 x 2 d x x 2 y = 3 x 3 3 + C x 2 y = x 3 + C y = x + C x 2 Now, include the point x = 0 in the domain. If x = 0 , the solution undefined except for C = 0 . That is, the equation has the unique solution y = x only when C = 0 . It is known that the unique solution for the IVP, d y d x + P ( x ) y = Q ( x ) , y ( x 0 ) = y 0 . is y ( x ) = 1 μ ( x ) [ ∫ μ ( x ) Q ( x ) d x + C ] where C = y 0 μ ( x 0 ) at the point x 0 . Therefore, the unique solution becomes y = x + y 0 μ ( x 0 ) x 2 . When y ( 0 ) = 0 , the only possible solution is y = x . When y ( 0 ) ≠ 0 , then C ≠ 0 and so y = x + C x 2 which is not defined at the point x = 0 . Thus, the equation has no solution. Thus, it is concluded that the uniqueness arrives from the fact that there are no constants in the solution. That is, y ( x ) = 1 μ ( x ) ∫ μ ( x ) Q ( x ) d x . Hence the required result is proved. (b) To show: That x y ′ − 2 y = 3 x has an infinite number of solutions defined at x = 0 . Then show that the initial value problem for this equation with initial condition y ( 0 ) = 0 has an infinite number of solutions. Consider the differential equation x y ′ − 2 y = 3 x Assume that the solution is defined at the point x = 0 and does not defined at the domain. Rewrite the differential equation as follows. x y ′ x − 2 y x = 3 x x y ′ − 2 x y = 3 d y d x − 2 x y = 3 ... ( 1 ) Here, P ( x ) = − 2 x and Q ( x ) = 3 . Calculate the integrating factor μ ( x ) by the formula μ ( x ) = exp [ ∫ P ( x ) d x ] . μ ( x ) = exp [ ∫ ( − 2 x ) d x ] = e − 2 ln x = e ln x − 2 = x − 2 = 1 x 2 Multiply the equation (1) by μ ( x ) and obtain d d x ( μ ( x ) y ) = μ ( x ) Q ( x ) . 1 x 2 ( d y d x − 2 x y ) = 3 1 x 2 1 x 2 d y d x − 2 x 3 y = 3 x 2 d d x ( y x 2 ) = 3 x 2 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 ( y x 2 ) = 3 x 2 and solve for y ( x ) by dividing μ ( x ) . y x 2 = ∫ 3 x 2 d x y x 2 = ∫ 3 x − 2 d x y x 2 = 3 x − 1 − 1 + C y = − 3 x + C x 2 Now, include the point x = 0 in the domain. If x = 0 , the solution defined for any choice of the constant C . Thus, the equation has infinite number of solutions. Substitute x = 0 in the solution y = − 3 x + C x 2 . y ( 0 ) = − 3 ( 0 ) + C ( 0 ) 2 = 0 Thus, for any choice of the constant C which satisfy the initial condition y ( 0 ) = 0 . Hence it is concluded that the initial value problem, y ′ − 2 x y = 3 , y ( 0 ) = 0 defined at x = 0 has infinite number of solutions for different values of the constant C .

(a) To show: That x y ′ + 2 y = 3 x has only one solution defined at x = 0 . Then show that the initial value problem for this equation with initial condition y ( 0 ) = y 0 has a unique solution when y 0 = 0 and no solution when y 0 ≠ 0 . Given: The initial value problem is, d y d x + P ( x ) y = x , y ( 0 ) = 1 where P ( x ) = { 1 , 0 ≤ x ≤ 2 , 3 , x > 2. Calculation: Consider the differential equation. x y ′ + 2 y = 3 x Assume that the solution is defined at the point x = 0 and does not defined at the domain. Rewrite the differential equation as follows. x y ′ x + 2 y x = 3 x x y ′ + 2 x y = 3 d y d x + 2 x y = 3 ... ( 1 ) Here, P ( x ) = 2 x and Q ( x ) = 3 . Calculate the integrating factor μ ( x ) by the formula μ ( x ) = exp [ ∫ P ( x ) d x ] . μ ( x ) = exp [ ∫ ( 2 x ) d x ] = e 2 ln x = e ln x 2 = x 2 Multiply the equation (1) by μ ( x ) and obtain d d x ( μ ( x ) y ) = μ ( x ) Q ( x ) . x 2 ( d y d x + 2 x y ) = 3 x 2 x 2 d y d x + 2 x 2 x y = 3 x 2 x 2 d y d x + 2 x y = 3 x 2 d d x ( x 2 y ) = 3 x 2 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 ( x 2 y ) = 3 x 2 and solve for y ( x ) by dividing μ ( x ) . x 2 y = ∫ 3 x 2 d x x 2 y = 3 x 3 3 + C x 2 y = x 3 + C y = x + C x 2 Now, include the point x = 0 in the domain. If x = 0 , the solution undefined except for C = 0 . That is, the equation has the unique solution y = x only when C = 0 . It is known that the unique solution for the IVP, d y d x + P ( x ) y = Q ( x ) , y ( x 0 ) = y 0 . is y ( x ) = 1 μ ( x ) [ ∫ μ ( x ) Q ( x ) d x + C ] where C = y 0 μ ( x 0 ) at the point x 0 . Therefore, the unique solution becomes y = x + y 0 μ ( x 0 ) x 2 . When y ( 0 ) = 0 , the only possible solution is y = x . When y ( 0 ) ≠ 0 , then C ≠ 0 and so y = x + C x 2 which is not defined at the point x = 0 . Thus, the equation has no solution. Thus, it is concluded that the uniqueness arrives from the fact that there are no constants in the solution. That is, y ( x ) = 1 μ ( x ) ∫ μ ( x ) Q ( x ) d x . Hence the required result is proved. (b) To show: That x y ′ − 2 y = 3 x has an infinite number of solutions defined at x = 0 . Then show that the initial value problem for this equation with initial condition y ( 0 ) = 0 has an infinite number of solutions. Consider the differential equation x y ′ − 2 y = 3 x Assume that the solution is defined at the point x = 0 and does not defined at the domain. Rewrite the differential equation as follows. x y ′ x − 2 y x = 3 x x y ′ − 2 x y = 3 d y d x − 2 x y = 3 ... ( 1 ) Here, P ( x ) = − 2 x and Q ( x ) = 3 . Calculate the integrating factor μ ( x ) by the formula μ ( x ) = exp [ ∫ P ( x ) d x ] . μ ( x ) = exp [ ∫ ( − 2 x ) d x ] = e − 2 ln x = e ln x − 2 = x − 2 = 1 x 2 Multiply the equation (1) by μ ( x ) and obtain d d x ( μ ( x ) y ) = μ ( x ) Q ( x ) . 1 x 2 ( d y d x − 2 x y ) = 3 1 x 2 1 x 2 d y d x − 2 x 3 y = 3 x 2 d d x ( y x 2 ) = 3 x 2 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 ( y x 2 ) = 3 x 2 and solve for y ( x ) by dividing μ ( x ) . y x 2 = ∫ 3 x 2 d x y x 2 = ∫ 3 x − 2 d x y x 2 = 3 x − 1 − 1 + C y = − 3 x + C x 2 Now, include the point x = 0 in the domain. If x = 0 , the solution defined for any choice of the constant C . Thus, the equation has infinite number of solutions. Substitute x = 0 in the solution y = − 3 x + C x 2 . y ( 0 ) = − 3 ( 0 ) + C ( 0 ) 2 = 0 Thus, for any choice of the constant C which satisfy the initial condition y ( 0 ) = 0 . Hence it is concluded that the initial value problem, y ′ − 2 x y = 3 , y ( 0 ) = 0 defined at x = 0 has infinite number of solutions for different values of the constant C .

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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Chapter 2.3, Problem 32E

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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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