Program Explanation Given information: The differential equation is ( x 2 − y 2 ) y ′ = 2 x y . Calculation: The given differential equation can be written as, d y d x = 2 x y ( x 2 − y 2 ) = 2 x y x 2 ( 1 − y 2 x 2 ) = 2 y x 1 − y 2 x 2 Substitute v = y x and differentiate with respect to x as, x v = y d d x ( x v ) = d y d x v d x d x + x d v d x = d y d x v + x d v d x = d y d x Now, the given differential equation can be written as, v + x d v d x = 2 v 1 − v 2 x d v d x = 2 v 1 − v 2 − v x d v d x = 2 v − v ( 1 − v 2 ) 1 − v 2 x d v d x = 2 v − v + v 3 1 − v 2 Simplify the above equation as, x d v d x = v + v 3 1 − v 2 1 − v 2 v + v 3 d v = d x x 1 − v 2 v ( 1 + v 2 ) d v = d x x Apply partial fraction decomposition to simplify the expression 1 − v 2 v ( 1 + v 2 ) as, 1 − v 2 v ( 1 + v 2 ) = A v + B v + D 1 + v 2 = A ( 1 + v 2 ) + v ( B v + D ) v ( 1 + v 2 ) </

5th Edition

ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Problems on Growing Functions

For a given set of functions f and g, the growth of functions states that the function g grows as fast as the function f. The growth rate of algorithms helps determine the algorithm's efficiency and compare its performance to other algorithms.

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Chapter 1.6, Problem 11P

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Program Description: Purpose of problem is to calculate the general solution of the differential equation (x2−y2)y′=2xy .

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Chapter 1.6, Problem 10P

Chapter 1.6, Problem 12P

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For the following function f and real number a, a. find the slope of the tangent line mtan = f' (a), and b. find the equation of the tangent line to f at x = a. f(x)= 2 = a = 2 x2 a. Slope: b. Equation of tangent line: y

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Chapter 1 Solutions

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Program Description: Purpose of problem is to calculate the general solution of the differential equation ( x 2 − y 2 ) y ′ = 2 x y .

Solution Summary: The author explains the purpose of the problem, which is to calculate the general solution of a differential equation.

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Program Description: Purpose of problem is to calculate the general solution of the differential equation (x2−y2)y′=2xy .

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Chapter 1 Solutions