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

The general solution to the equation d x d t + t x 3 + x t = 0 . The general solution of the differential equation d x d t + t x 3 + x t = 0 is x 2 = 1 2 t 2 ln | t | + C t 2 and x ≡ 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 x d t + t x 3 + x t d x d t + x t = − t x 3 It is observed that the given equation is a Bernoulli equation with n = 3 , P ( t ) = 1 t and Q ( t ) = − t . First divide the equation by x 3 , to transform the equation into a linear equation as follows. 1 x 3 d x d t + x x 3 t = − t x 3 x 3 1 x 3 d x d t + 1 x 2 t = − t x − 3 d x d t + 1 t x − 2 = − t Next make the substitution v = x − 2 and differentiate it implicitly. Since d v d t = − 2 x − 3 d x d t , the transformed equation is − 1 2 d v d t + v t = − t d v d t − 2 t v = 2 t ... ( 1 ) This is a linear equation of the standard form d x d t + P ( t ) x = Q ( t ) . Here, P ( t ) = − 2 t and Q ( t ) = 2 t . Calculate the integrating factor μ ( t ) by the formula μ ( t ) = exp [ ∫ P ( t ) d t ] . μ ( t ) = exp [ ∫ ( − 2 t ) d t ] = exp [ − 2 ln t ] = e ln t − 2 = t − 2 Multiply the equation (1) by μ ( t ) and obtain d d t ( μ ( t ) v ) = μ ( t ) Q ( t ) . t − 2 ( d v d t − 2 t v ) = t − 2 2 t t − 2 d v d t − 2 t − 3 v = 2 t − 1 d d t ( t − 2 v ) = 2 t To obtain the general solution v ( t ) = 1 μ ( t ) [ ∫ μ ( t ) Q ( t ) d t + C ] , integrate both sides of the equation d d t ( t − 2 v ) = 2 t and solve for v ( t ) by dividing μ ( t ) . t − 2 v = ∫ 2 t d t t − 2 v = 2 ln | t | + C v = 2 ln | t | t − 2 + C t − 2 v = 2 t 2 ln | t | + C t 2 Substitute back, v = x − 2 in the above equation. x − 2 = 2 t 2 ln | t | + C t 2 1 x 2 = 2 t 2 ln | t | + C t 2 x 2 = 1 2 t 2 ln | t | + C t 2 where x ≠ 0 ( ∵ the equation divided by x 3 in the first step ) Another solution is x ≡ 0 . Thus, the general solution of the differential equation d x d t + t x 3 + x t = 0 is x 2 = 1 2 t 2 ln | t | + C t 2 and x ≡ 0 _ .

The general solution to the equation d x d t + t x 3 + x t = 0 . The general solution of the differential equation d x d t + t x 3 + x t = 0 is x 2 = 1 2 t 2 ln | t | + C t 2 and x ≡ 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 x d t + t x 3 + x t d x d t + x t = − t x 3 It is observed that the given equation is a Bernoulli equation with n = 3 , P ( t ) = 1 t and Q ( t ) = − t . First divide the equation by x 3 , to transform the equation into a linear equation as follows. 1 x 3 d x d t + x x 3 t = − t x 3 x 3 1 x 3 d x d t + 1 x 2 t = − t x − 3 d x d t + 1 t x − 2 = − t Next make the substitution v = x − 2 and differentiate it implicitly. Since d v d t = − 2 x − 3 d x d t , the transformed equation is − 1 2 d v d t + v t = − t d v d t − 2 t v = 2 t ... ( 1 ) This is a linear equation of the standard form d x d t + P ( t ) x = Q ( t ) . Here, P ( t ) = − 2 t and Q ( t ) = 2 t . Calculate the integrating factor μ ( t ) by the formula μ ( t ) = exp [ ∫ P ( t ) d t ] . μ ( t ) = exp [ ∫ ( − 2 t ) d t ] = exp [ − 2 ln t ] = e ln t − 2 = t − 2 Multiply the equation (1) by μ ( t ) and obtain d d t ( μ ( t ) v ) = μ ( t ) Q ( t ) . t − 2 ( d v d t − 2 t v ) = t − 2 2 t t − 2 d v d t − 2 t − 3 v = 2 t − 1 d d t ( t − 2 v ) = 2 t To obtain the general solution v ( t ) = 1 μ ( t ) [ ∫ μ ( t ) Q ( t ) d t + C ] , integrate both sides of the equation d d t ( t − 2 v ) = 2 t and solve for v ( t ) by dividing μ ( t ) . t − 2 v = ∫ 2 t d t t − 2 v = 2 ln | t | + C v = 2 ln | t | t − 2 + C t − 2 v = 2 t 2 ln | t | + C t 2 Substitute back, v = x − 2 in the above equation. x − 2 = 2 t 2 ln | t | + C t 2 1 x 2 = 2 t 2 ln | t | + C t 2 x 2 = 1 2 t 2 ln | t | + C t 2 where x ≠ 0 ( ∵ the equation divided by x 3 in the first step ) Another solution is x ≡ 0 . Thus, the general solution of the differential equation d x d t + t x 3 + x t = 0 is x 2 = 1 2 t 2 ln | t | + C t 2 and x ≡ 0 _ .

Solution Summary: The author explains how the general solution of the differential equation dxt=0 is a 'Bernoulli equation'.

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. (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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