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

5th Edition

ISBN: 8220100802454

Author: Calvis

Publisher: PEARSON

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Rod Cutting

A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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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

EBK DIFFERENTIAL EQUATIONS

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Show that the substitution...Ch. 1.6 - Prob. 57P Ch. 1.6 - Prob. 58P Ch. 1.6 - Solve the differential equation dydx=xy1x+y+3 by...Ch. 1.6 - Prob. 60P Ch. 1.6 - Prob. 61P Ch. 1.6 - Prob. 62P Ch. 1.6 - Prob. 63P Ch. 1.6 - Prob. 64P Ch. 1.6 - Prob. 65P Ch. 1.6 - Prob. 66P Ch. 1.6 - Prob. 67P Ch. 1.6 - Prob. 68P Ch. 1.6 - Prob. 69P Ch. 1.6 - As in the text discussion, suppose that an...Ch. 1.6 - Prob. 71P Ch. 1.6 - Prob. 72P Ch. 1 - Prob. 1RP Ch. 1 - Prob. 2RP Ch. 1 - Prob. 3RP Ch. 1 - Prob. 4RP Ch. 1 - Prob. 5RP Ch. 1 - Prob. 6RP Ch. 1 - Prob. 7RP Ch. 1 - Prob. 8RP Ch. 1 - Prob. 9RP Ch. 1 - Prob. 10RP Ch. 1 - Prob. 11RP Ch. 1 - Prob. 12RP Ch. 1 - Prob. 13RP Ch. 1 - Prob. 14RP Ch. 1 - Prob. 15RP Ch. 1 - Prob. 16RP Ch. 1 - Prob. 17RP Ch. 1 - Prob. 18RP Ch. 1 - Prob. 19RP Ch. 1 - Prob. 20RP Ch. 1 - Prob. 21RP Ch. 1 - Prob. 22RP Ch. 1 - Prob. 23RP Ch. 1 - Prob. 24RP Ch. 1 - Prob. 25RP Ch. 1 - Prob. 26RP Ch. 1 - Prob. 27RP Ch. 1 - Prob. 28RP Ch. 1 - Prob. 29RP Ch. 1 - Prob. 30RP Ch. 1 - Prob. 31RP Ch. 1 - Prob. 32RP Ch. 1 - Prob. 33RP Ch. 1 - Prob. 34RP Ch. 1 - Prob. 35RP Ch. 1 - Prob. 36RP

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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.

Concept explainers

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

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Chapter 1 Solutions