Program Explanation Given information: The reducible second-order differential equation is, y 2 y ′ + 2 x y 3 = 6 x Explanation: The given differential equation can be rewritten as, d y d x + 2 x y 3 y 2 = 6 x y 2 d y d x + 2 x y = 6 x y − 2 Compare the Bernoulli equation with the above equation to obtain the value of P ( x ) and Q ( x ) . P ( x ) = 2 x Q ( x ) = 6 x The value of n is − 2 and the value of 1 − n is calculated as, 1 − n = 1 − ( − 2 ) = 1 + 2 = 3 Substitute y 3 for v and differentiate with respect to x as, v = y 3 y = v 1 3 d y d x = 1 3 v − 2 3 d v d x The given differential equation can be written as, 1 3 v − 2 3 d v d x + 2 x v 1 3 = 6 x ( v 1 3 ) − 2 1 3 v − 2 3 d v d x + 2 x v 1 3 = 6 x v − 2 3 1 3 v − 2 3 ( d v d x + 6 x v ) = 6 x v − 2 3 d v d x + 6 x v = 18 x The above equation is a linear first-order equation and its solutio

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Question

Chapter 1.6, Problem 20P

Program Plan Intro

Program Description: Purpose ofproblem is to calculate the general solution of the differential equation y2y′+2xy3=6x .

Summary introduction: Problem will use the Bernoulli equation of the form dydx+P(x)y=Q(x)yn and then the substitution v=y1−n .

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Chapter 1.6, Problem 19P

Chapter 1.6, Problem 21P

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Chapter 1 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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Program Description: Purpose ofproblem is to calculate the general solution of the differential equation y 2 y ′ + 2 x y 3 = 6 x . Summary introduction: Problem will use the Bernoulli equation of the form d y d x + P ( x ) y = Q ( x ) y n and then the substitution v = y 1 − n .

Solution Summary: The author explains that the problem uses the Bernoulli equation and compares it with the above equation to obtain the values of P(x) and 1-n

Concept explainers

Program Description: Purpose ofproblem is to calculate the general solution of the differential equation y2y′+2xy3=6x .

Summary introduction: Problem will use the Bernoulli equation of the form dydx+P(x)y=Q(x)yn and then the substitution v=y1−n .

Want to see the full answer?

Chapter 1 Solutions

Program Description: Purpose ofproblem is to calculate the general solution of the differential equation y 2 y ′ + 2 x y 3 = 6 x . Summary introduction: Problem will use the Bernoulli equation of the form d y d x + P ( x ) y = Q ( x ) y n and then the substitution v = y 1 − n .

Solution Summary: The author explains that the problem uses the Bernoulli equation and compares it with the above equation to obtain the values of P(x) and 1-n

Concept explainers

Program Description: Purpose ofproblem is to calculate the general solution of the differential equation y2y′+2xy3=6x . Summary introduction: Problem will use the Bernoulli equation of the form dydx+P(x)y=Q(x)yn and then the substitution v=y1−n .

Want to see the full answer?

Chapter 1 Solutions

Program Description: Purpose ofproblem is to calculate the general solution of the differential equation y2y′+2xy3=6x .

Summary introduction: Problem will use the Bernoulli equation of the form dydx+P(x)y=Q(x)yn and then the substitution v=y1−n .