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

The given equation ( y − 4 x − 1 ) 2 d x − d y = 0 in Problem 2. The solution of the differential equation ( y − 4 x − 1 ) 2 d x − d y = 0 is y ( x ) = 4 x + 3 − 4 k x e 4 x + k e 4 x 1 − k e 4 x _ . Definition used: “The equation is of the form d y d x = G ( a x + b y ) . Let z = a x + b y . Then d z d x = a + b ( d y d x ) , and the transformed equation in the variables z and x is separable.” Calculation: The differential equation is, ( y − 4 x − 1 ) 2 d x − d y = 0 Rewrite the equation ( y − 4 x − 1 ) 2 d x − d y = 0 as shown below. ( y − 4 x − 1 ) 2 d x − d y = 0 − d y = − ( y − 4 x − 1 ) 2 d x d y d x = ( y − 4 x − 1 ) 2 The above equation is of the form d y d x = G ( x + y ) . So, take z = y − 4 x − 1 . Then d y d x = d z d x + 4 , the transformed equation is, d z d x + 4 = z 2 d z d x = z 2 − 4 Separate the variables and rewrite the differential equation as follows. 1 z 2 − 4 d z = d x Integrate on both sides of the equation. ∫ 1 z 2 − 4 d z = ∫ d x ∫ 1 z 2 − 4 d z = x + C 1 ∫ 1 z 2 − 2 2 d z = x + C 1 1 4 ln ( z − 2 z + 2 ) = x + C 1 ( ∵ ∫ 1 x 2 − a 2 d x = 1 2 a ln ( x − a x + a ) ) Simplify further as follows. ln ( z − 2 z + 2 ) = 4 x + 4 C 1 ln ( z − 2 z + 2 ) = e 4 x + 4 C 1 z − 2 z + 2 = k e 4 x ( Take k = e 4 C 1 ) Substitute back, z = y − 4 x − 1 . y − 4 x − 1 − 2 y − 4 x − 1 + 2 = k e 4 x y − 4 x − 3 y − 4 x + 1 = k e 4 x Simplify further as follows. y − 4 x − 3 = k e 4 x ( y − 4 x + 1 ) y − 4 x − 3 = k e 4 x y − 4 x k e 4 x + k e 4 x y − k e 4 x y = 4 x + 3 − 4 x k e 4 x + k e 4 x y ( x ) = 4 x + 3 − 4 k x e 4 x + k e 4 x 1 − k e 4 x Thus, the solution of the differential equation ( y − 4 x − 1 ) 2 d x − d y = 0 is y ( x ) = 4 x + 3 − 4 k x e 4 x + k e 4 x 1 − k e 4 x _ .

The given equation ( y − 4 x − 1 ) 2 d x − d y = 0 in Problem 2. The solution of the differential equation ( y − 4 x − 1 ) 2 d x − d y = 0 is y ( x ) = 4 x + 3 − 4 k x e 4 x + k e 4 x 1 − k e 4 x _ . Definition used: “The equation is of the form d y d x = G ( a x + b y ) . Let z = a x + b y . Then d z d x = a + b ( d y d x ) , and the transformed equation in the variables z and x is separable.” Calculation: The differential equation is, ( y − 4 x − 1 ) 2 d x − d y = 0 Rewrite the equation ( y − 4 x − 1 ) 2 d x − d y = 0 as shown below. ( y − 4 x − 1 ) 2 d x − d y = 0 − d y = − ( y − 4 x − 1 ) 2 d x d y d x = ( y − 4 x − 1 ) 2 The above equation is of the form d y d x = G ( x + y ) . So, take z = y − 4 x − 1 . Then d y d x = d z d x + 4 , the transformed equation is, d z d x + 4 = z 2 d z d x = z 2 − 4 Separate the variables and rewrite the differential equation as follows. 1 z 2 − 4 d z = d x Integrate on both sides of the equation. ∫ 1 z 2 − 4 d z = ∫ d x ∫ 1 z 2 − 4 d z = x + C 1 ∫ 1 z 2 − 2 2 d z = x + C 1 1 4 ln ( z − 2 z + 2 ) = x + C 1 ( ∵ ∫ 1 x 2 − a 2 d x = 1 2 a ln ( x − a x + a ) ) Simplify further as follows. ln ( z − 2 z + 2 ) = 4 x + 4 C 1 ln ( z − 2 z + 2 ) = e 4 x + 4 C 1 z − 2 z + 2 = k e 4 x ( Take k = e 4 C 1 ) Substitute back, z = y − 4 x − 1 . y − 4 x − 1 − 2 y − 4 x − 1 + 2 = k e 4 x y − 4 x − 3 y − 4 x + 1 = k e 4 x Simplify further as follows. y − 4 x − 3 = k e 4 x ( y − 4 x + 1 ) y − 4 x − 3 = k e 4 x y − 4 x k e 4 x + k e 4 x y − k e 4 x y = 4 x + 3 − 4 x k e 4 x + k e 4 x y ( x ) = 4 x + 3 − 4 k x e 4 x + k e 4 x 1 − k e 4 x Thus, the solution of the differential equation ( y − 4 x − 1 ) 2 d x − d y = 0 is y ( x ) = 4 x + 3 − 4 k x e 4 x + k e 4 x 1 − k e 4 x _ .

Solution Summary: The author explains the solution of the differential equation (y-4x-1)2dx-dy=0.

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 2.6, Problem 33E

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)5. The operator T on a vector space V over Q has as its sequence of invariant factors the sequence (x-1)3(x-2)5(x-3)2, (x-1)(x-2)(x-3)2, (x-3). Find the Jordan normal form of T.

18

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. 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