Program Explanation Given information: The differential equation is, d y d x + y 2 = 1 + x 2 One particular solution of Riccati equation is y 1 ( x ) = x . Explanation: The Riccati equation is, d y d x = A ( x ) y 2 + B ( x ) y + C ( x ) The given differential equation can be written as, d y d x = 1 + x 2 − y 2 Consider x + 1 v as y and differentiate with respect to x . 1 − 1 v 2 d v d x = d y d x Substitute 1 − 1 v 2 d v d x for d y d x in given differential equation. 1 − 1 v 2 d v d x = 1 + x 2 − y 2 − 1 v 2 d v d x = x 2 − y 2 1 v 2 d v d x = y 2 − x 2 = ( y − x ) ( y + x ) The value of y − x is calculated as, y = x + 1 v y − x = 1 v The value of y + x is calculated as, y = x + 1 v y + x = 2 x + 1 v Now, the given differential equation can be written as, 1 v 2 d v d x = 1 v ( 2 x + 1 v ) = 2 x v + 1 v 2 d v d x = 2 x v + 1 d v d x − 2 x v = 1 The above equation is a linear first-order equation and its solution is calculated as, v e − 2 ∫ x d x = ∫ e − 2 ∫ x d x d x + C v e − 2 ⋅ x 2 2 = ∫ e − 2 ⋅ x 2 2 d x + C v e − x 2 = ∫ e − x 2 d x + C Simplify the above equation

MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- For Differential Equations and Boundary Value Problems: Computing and Modeling Tech Update

5th Edition

ISBN: 9780134872971

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

Publisher: PEARSON

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Chapter 1.6, Problem 64P

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Program Description: Purpose ofproblem is to calculate solution of the differential equation dydx+y2=1+x2 .

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Chapter 1.6, Problem 63P

Chapter 1.6, Problem 65P

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Chapter 1 Solutions

MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- For Differential Equations and Boundary Value Problems: Computing and Modeling Tech Update

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Program Description: Purpose ofproblem is to calculate solution of the differential equation d y d x + y 2 = 1 + x 2 .

Solution Summary: The author explains that the Riccati equation is y_1(x)=x.

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Program Description: Purpose ofproblem is to calculate solution of the differential equation dydx+y2=1+x2 .

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Chapter 1 Solutions