(a) Explanation Given: The Mathematical model is, x v d v d x + v 2 = 32 x . Here, velocity of chain is v and the length of chain at an instant is x . Initial length of the chain is 3 ft and the initial velocity is 0 (rest). Procedure used: Compare the given differential equation with the standard form of exact differential equation M d x + N d v = 0 . To check whether the given differential equation is exact or not, check the condition ∂ M ∂ v = ∂ N ∂ x is satisfied or not. If the given differential equation is not exact, then make it exact by using an appropriate integrating factor. Calculate ∂ M ∂ v − ∂ N ∂ x N , if this term is a function of x only or a constant then integrating factor is e ∫ f ( x ) d x . Standard solution of the exact differential equation: ∫ M d x + ∫ ( t e r m s o f N f r e e f r o m x ) d v = C Calculation: Simplify the given mathematical model as, x v d v d x + v 2 = 32 x x v d v d x = 32 x − v 2 x v d v = ( 32 x − v 2 ) d x ( v 2 − 32 x ) d x + x v d v = 0 (1) Compare equation (1) with the standard form of exact differential equation. Thus, M = v 2 − 32 and N = x v . Calculate ∂ M ∂ v as follows. ∂ M ∂ v = ∂ ∂ v ( v 2 − 32 x ) = 2 v Calculate ∂ N ∂ x as follows. ∂ N ∂ x = ∂ ∂ x ( x v ) = v Since ∂ M ∂ y ≠ ∂ N ∂ x , the given differential equation is not exact. Calculate ∂ M ∂ y − ∂ N ∂ x N , by substituting 2 v for ∂ M ∂ v , v for ∂ N ∂ x and x v for N . ∂ M ∂ v − ∂ N ∂ x N = 2 v − v x v = v x v = 1 x Since the value of ∂ M ∂ v − ∂ N ∂ x N is the function of x only, integrating factor for the given differential equation is, I F = e ∫ 1 x d x = e ln x = x Multiply equation (1) by the integrating factor (b) To determine The velocity by which the chain leaves the platform.

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

See similar textbooks

Videos

Question

Chapter 2.4, Problem 45E

(a)

To determine

An explicit solution for the given mathematical model.

(b)

To determine

The velocity by which the chain leaves the platform.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.4, Problem 44E

Chapter 2.5, Problem 1E

Chapter 2 Solutions

Advanced Engineering Mathematics

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.1 - Prob. 37E Ch. 2.1 - Prob. 38E Ch. 2.1 - Prob. 39E Ch. 2.1 - Prob. 40E Ch. 2.1 - Prob. 41E Ch. 2.1 - Prob. 42E Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - Prob. 13E Ch. 2.2 - Prob. 14E Ch. 2.2 - Prob. 15E Ch. 2.2 - Prob. 16E Ch. 2.2 - Prob. 17E Ch. 2.2 - Prob. 18E Ch. 2.2 - Prob. 19E Ch. 2.2 - Prob. 20E Ch. 2.2 - Prob. 21E Ch. 2.2 - Prob. 22E Ch. 2.2 - Prob. 23E Ch. 2.2 - Prob. 24E Ch. 2.2 - Prob. 25E Ch. 2.2 - Prob. 26E Ch. 2.2 - Prob. 27E Ch. 2.2 - Prob. 28E Ch. 2.2 - Prob. 29E Ch. 2.2 - Prob. 30E Ch. 2.2 - Prob. 31E Ch. 2.2 - Prob. 32E Ch. 2.2 - Prob. 33E Ch. 2.2 - Prob. 34E Ch. 2.2 - Prob. 35E Ch. 2.2 - Prob. 36E Ch. 2.2 - Prob. 37E Ch. 2.2 - Prob. 38E Ch. 2.2 - Prob. 39E Ch. 2.2 - Prob. 40E Ch. 2.2 - Prob. 41E Ch. 2.2 - Prob. 42E Ch. 2.2 - Prob. 43E Ch. 2.2 - Prob. 44E Ch. 2.2 - Prob. 45E Ch. 2.2 - Prob. 46E Ch. 2.2 - Prob. 47E Ch. 2.2 - Prob. 48E Ch. 2.2 - Prob. 49E Ch. 2.2 - Prob. 50E Ch. 2.2 - Prob. 51E Ch. 2.2 - Prob. 52E Ch. 2.2 - Prob. 53E Ch. 2.3 - Prob. 1E Ch. 2.3 - Prob. 2E Ch. 2.3 - Prob. 3E Ch. 2.3 - Prob. 4E Ch. 2.3 - Prob. 5E Ch. 2.3 - Prob. 6E Ch. 2.3 - Prob. 7E Ch. 2.3 - Prob. 8E Ch. 2.3 - Prob. 9E Ch. 2.3 - Prob. 10E Ch. 2.3 - Prob. 11E Ch. 2.3 - Prob. 12E Ch. 2.3 - Prob. 13E Ch. 2.3 - Prob. 14E Ch. 2.3 - Prob. 15E Ch. 2.3 - Prob. 16E Ch. 2.3 - Prob. 17E Ch. 2.3 - Prob. 18E Ch. 2.3 - Prob. 19E Ch. 2.3 - Prob. 20E Ch. 2.3 - Prob. 21E Ch. 2.3 - Prob. 22E Ch. 2.3 - Prob. 23E Ch. 2.3 - Prob. 24E Ch. 2.3 - Prob. 25E Ch. 2.3 - Prob. 26E Ch. 2.3 - Prob. 27E Ch. 2.3 - Prob. 28E Ch. 2.3 - Prob. 29E Ch. 2.3 - Prob. 30E 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. 2.3 - Prob. 40E Ch. 2.3 - Prob. 41E Ch. 2.3 - Prob. 42E Ch. 2.3 - Prob. 43E Ch. 2.3 - Prob. 44E Ch. 2.3 - Prob. 45E Ch. 2.3 - Prob. 46E Ch. 2.3 - Prob. 47E Ch. 2.3 - Prob. 48E Ch. 2.3 - Prob. 49E Ch. 2.3 - Prob. 50E Ch. 2.3 - Prob. 51E Ch. 2.3 - Prob. 52E Ch. 2.3 - Prob. 53E Ch. 2.3 - Prob. 56E Ch. 2.4 - Prob. 1E Ch. 2.4 - Prob. 2E Ch. 2.4 - Prob. 3E Ch. 2.4 - Prob. 4E Ch. 2.4 - Prob. 5E Ch. 2.4 - Prob. 6E Ch. 2.4 - Prob. 7E Ch. 2.4 - Prob. 8E Ch. 2.4 - Prob. 9E Ch. 2.4 - Prob. 10E Ch. 2.4 - Prob. 11E Ch. 2.4 - Prob. 12E Ch. 2.4 - Prob. 13E Ch. 2.4 - Prob. 14E Ch. 2.4 - Prob. 15E Ch. 2.4 - Prob. 16E Ch. 2.4 - Prob. 17E Ch. 2.4 - Prob. 18E Ch. 2.4 - Prob. 19E Ch. 2.4 - Prob. 20E Ch. 2.4 - Prob. 21E Ch. 2.4 - Prob. 22E Ch. 2.4 - Prob. 23E Ch. 2.4 - Prob. 24E Ch. 2.4 - Prob. 25E Ch. 2.4 - Prob. 26E Ch. 2.4 - Prob. 27E Ch. 2.4 - Prob. 28E Ch. 2.4 - Prob. 29E Ch. 2.4 - Prob. 30E Ch. 2.4 - Prob. 31E Ch. 2.4 - Prob. 32E Ch. 2.4 - Prob. 33E Ch. 2.4 - Prob. 34E Ch. 2.4 - Prob. 35E Ch. 2.4 - Prob. 36E Ch. 2.4 - Prob. 37E Ch. 2.4 - Prob. 38E Ch. 2.4 - Prob. 39E Ch. 2.4 - Prob. 40E Ch. 2.4 - Prob. 41E Ch. 2.4 - Prob. 42E Ch. 2.4 - Prob. 43E Ch. 2.4 - Prob. 44E Ch. 2.4 - Prob. 45E Ch. 2.5 - Prob. 1E Ch. 2.5 - Prob. 2E Ch. 2.5 - Prob. 3E Ch. 2.5 - Prob. 4E Ch. 2.5 - Prob. 5E 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 - Prob. 20E Ch. 2.5 - Prob. 21E Ch. 2.5 - Prob. 22E Ch. 2.5 - Prob. 23E Ch. 2.5 - Prob. 24E Ch. 2.5 - Prob. 25E Ch. 2.5 - Prob. 26E Ch. 2.5 - Prob. 27E Ch. 2.5 - Prob. 28E Ch. 2.5 - Prob. 29E Ch. 2.5 - Prob. 30E Ch. 2.5 - Prob. 31E Ch. 2.5 - Prob. 32E Ch. 2.5 - Prob. 33E Ch. 2.5 - Prob. 34E Ch. 2.5 - Prob. 35E Ch. 2.5 - Prob. 36E Ch. 2.5 - Prob. 37E Ch. 2.6 - Prob. 1E 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 - Prob. 7E Ch. 2.6 - Prob. 8E Ch. 2.6 - Prob. 9E Ch. 2.6 - Prob. 10E Ch. 2.7 - Prob. 1E Ch. 2.7 - Prob. 2E Ch. 2.7 - Prob. 3E Ch. 2.7 - Prob. 4E Ch. 2.7 - Prob. 5E Ch. 2.7 - Prob. 6E Ch. 2.7 - Prob. 7E Ch. 2.7 - Prob. 8E Ch. 2.7 - Prob. 9E Ch. 2.7 - Prob. 10E Ch. 2.7 - Prob. 11E Ch. 2.7 - Prob. 12E Ch. 2.7 - Prob. 13E Ch. 2.7 - Prob. 14E Ch. 2.7 - Prob. 15E Ch. 2.7 - Prob. 16E Ch. 2.7 - Prob. 17E Ch. 2.7 - Prob. 18E Ch. 2.7 - Prob. 19E Ch. 2.7 - Prob. 20E Ch. 2.7 - Prob. 21E Ch. 2.7 - Prob. 22E Ch. 2.7 - Prob. 23E Ch. 2.7 - Prob. 24E Ch. 2.7 - Prob. 25E Ch. 2.7 - Prob. 26E Ch. 2.7 - Prob. 27E Ch. 2.7 - Prob. 28E Ch. 2.7 - Prob. 29E Ch. 2.7 - Prob. 30E Ch. 2.7 - Prob. 31E Ch. 2.7 - Prob. 32E Ch. 2.7 - Prob. 33E Ch. 2.7 - Prob. 34E Ch. 2.7 - Prob. 35E Ch. 2.7 - Prob. 36E Ch. 2.7 - Prob. 37E Ch. 2.7 - Prob. 38E Ch. 2.7 - Prob. 39E Ch. 2.7 - Prob. 40E Ch. 2.7 - Prob. 41E Ch. 2.7 - Prob. 42E Ch. 2.7 - Prob. 43E Ch. 2.7 - Prob. 44E Ch. 2.7 - Prob. 45E Ch. 2.7 - Prob. 46E Ch. 2.7 - Prob. 48E Ch. 2.7 - Prob. 50E Ch. 2.7 - Prob. 51E Ch. 2.8 - Prob. 1E Ch. 2.8 - Prob. 2E Ch. 2.8 - Prob. 3E Ch. 2.8 - Prob. 4E Ch. 2.8 - Prob. 5E Ch. 2.8 - Prob. 6E Ch. 2.8 - Prob. 7E Ch. 2.8 - Prob. 8E Ch. 2.8 - Prob. 9E Ch. 2.8 - Prob. 10E Ch. 2.8 - Prob. 11E Ch. 2.8 - Prob. 12E Ch. 2.8 - Prob. 13E Ch. 2.8 - Prob. 14E Ch. 2.8 - Prob. 15E Ch. 2.8 - Prob. 16E Ch. 2.8 - Prob. 17E Ch. 2.8 - Prob. 18E Ch. 2.8 - Prob. 19E Ch. 2.8 - Prob. 20E Ch. 2.8 - Prob. 21E Ch. 2.8 - Prob. 22E Ch. 2.8 - Prob. 23E Ch. 2.8 - Prob. 24E Ch. 2.8 - Prob. 30E Ch. 2.9 - Prob. 1E Ch. 2.9 - Prob. 2E Ch. 2.9 - Prob. 3E Ch. 2.9 - Prob. 4E Ch. 2.9 - Prob. 7E Ch. 2.9 - Prob. 8E Ch. 2.9 - Prob. 9E Ch. 2.9 - Prob. 10E Ch. 2.9 - Prob. 11E Ch. 2.9 - Prob. 14E Ch. 2.9 - Prob. 15E Ch. 2.9 - Prob. 16E Ch. 2.9 - Prob. 17E Ch. 2.9 - Prob. 18E Ch. 2 - Prob. 1CR Ch. 2 - Prob. 2CR Ch. 2 - Prob. 3CR Ch. 2 - Prob. 4CR Ch. 2 - Prob. 5CR Ch. 2 - Prob. 6CR Ch. 2 - Prob. 7CR Ch. 2 - Prob. 8CR Ch. 2 - Prob. 9CR Ch. 2 - Prob. 10CR Ch. 2 - Prob. 11CR Ch. 2 - Prob. 12CR Ch. 2 - Prob. 13CR Ch. 2 - Prob. 14CR Ch. 2 - Prob. 15CR Ch. 2 - Prob. 16CR Ch. 2 - Prob. 17CR Ch. 2 - Prob. 18CR Ch. 2 - Prob. 19CR Ch. 2 - Prob. 20CR Ch. 2 - Prob. 21CR Ch. 2 - Prob. 22CR Ch. 2 - Prob. 23CR Ch. 2 - Prob. 24CR Ch. 2 - Prob. 25CR Ch. 2 - Prob. 26CR Ch. 2 - Prob. 27CR Ch. 2 - Prob. 28CR Ch. 2 - Prob. 29CR Ch. 2 - Prob. 30CR Ch. 2 - Prob. 31CR Ch. 2 - Prob. 32CR Ch. 2 - Prob. 33CR Ch. 2 - Prob. 34CR Ch. 2 - Prob. 35CR Ch. 2 - Prob. 36CR Ch. 2 - Prob. 37CR Ch. 2 - Prob. 38CR Ch. 2 - Prob. 39CR Ch. 2 - Prob. 40CR Ch. 2 - Prob. 41CR Ch. 2 - Prob. 42CR Ch. 2 - Prob. 43CR Ch. 2 - Prob. 44CR Ch. 2 - Prob. 45CR Ch. 2 - Prob. 46CR Ch. 2 - Prob. 47CR Ch. 2 - Prob. 48CR Ch. 2 - Prob. 49CR Ch. 2 - Prob. 50CR

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

An explicit solution for the given mathematical model.

Solution Summary: The author explains how to simplify the given mathematical model by comparing it with the standard form of exact differential equation Mdx+Ndv.

Videos

An explicit solution for the given mathematical model.

The velocity by which the chain leaves the platform.

Want to see the full answer?

Chapter 2 Solutions