Program Explanation Given information: The differential equation is d y d x = 2 x y + 2 x x 2 + 1 . Explanation: The given differential equation can be solved by separable variable method and linear from method. The given differential equation can be written as, d y d x = 2 x ( y + 1 ) x 2 + 1 d y y + 1 = 2 x x 2 + 1 d x The above equation is in separable form and it can be solved as, ∫ d y y + 1 = ∫ 2 x x 2 + 1 d x ln ( y + 1 ) = ln ( x 2 + 1 ) + ln C ln ( y + 1 ) = ln C ( x 2 + 1 ) y + 1 = C ( x 2 + 1 ) Simplify the above equation. y = C ( x 2 + 1 ) − 1 Here, C is constant of integration. The given differential equation can also be written as, d y d x = 2 x y x 2 + 1 + 2 x x 2 + 1 d y d x − 2 x x 2 + 1 y = 2 x x 2 + 1 The above equation is in form of linear first-order equation and its solution is calculated as, y e − ∫ 2 x x 2 + 1 d x = ∫ 2 x x 2 + 1 e − ∫ 2 x x 2 + 1 d x d x y e − ln ( x 2 + 1

5th Edition

ISBN: 9780135962176

Author: Edwards

Publisher: PEARSON CO

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Chapter 1, Problem 35RP

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Program Description: Purpose of problem is to derive general solutions of the differential equation dydx=2xy+2xx2+1 in two different ways.

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Chapter 1, Problem 34RP

Chapter 1, Problem 36RP

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Question 1: If a barometer were built using oil (p = 0.92 g/cm³) instead of mercury (p = 13.6 g/cm³), would the column of oil be higher than, lower than, or the same as the column of mercury at 1.00 atm? If the level is different, by what factor? Explain. (5 pts) Solution: A barometer works based on the principle that the pressure exerted by the liquid column balances atmospheric pressure. The pressure is given by: P = pgh Since the atmospheric pressure remains constant (P = 1.00 atm), the height of the liquid column is inversely proportional to its density: Step 1: Given Data PHg hol=hgx Poil • Density of mercury: PHg = 13.6 g/cm³ Density of oil: Poil = 0.92 g/cm³ • Standard height of mercury at 1.00 atm: hμg Step 2: Compute Height of Oil = 760 mm = 0.760 m 13.6 hoil = 0.760 x 0.92 hoil = 0.760 × 14.78 hoil = 11.23 m Step 3: Compare Heights Since oil is less dense than mercury, the column of oil must be much taller than that of mercury. The factor by which it is taller is: Final…

Question 3: A sealed flask at room temperature contains a mixture of neon (Ne) and nitrogen (N2) gases. Ne has a mass of 3.25 g and exerts a pressure of 48.2 torr. . N2 contributes a pressure of 142 torr. • What is the mass of the N2 in the flask? • Atomic mass of Ne = 20.1797 g/mol • Atomic mass of N = 14.0067 g/mol Solution: We will use the Ideal Gas Law to determine the number of moles of each gas and calculate the mass of N2. PV = nRT where: • P = total pressure • V volume of the flask (same for both gases) n = number of moles of gas • R 0.0821 L atm/mol K • T = Room temperature (assume 298 K) Since both gases are in the same flask, their partial pressures correspond to their mole fractions. Step 1: Convert Pressures to Atmospheres 48.2 PNe = 0.0634 atm 760 142 PN2 = = 0.1868 atm 760 Step 2: Determine Moles of Ne nNe = mass molar mass 3.25 nNe 20.1797 nne 0.1611 mol Step 3: Use Partial Pressure Ratio to Find n

Chapter 1 Solutions

EP DIFFERENTIAL EQUATIONS+..-MYLAB ACCE

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Program Description: Purpose of problem is to derive general solutions of the differential equation d y d x = 2 x y + 2 x x 2 + 1 in two different ways.

Solution Summary: The author explains that the given differential equation can be solved by separable variable method and linear from method.

Program Description: Purpose of problem is to derive general solutions of the differential equation dydx=2xy+2xx2+1 in two different ways.

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Chapter 1 Solutions