Explanation Calculation: Consider the given equation. ( x 2 + y 2 ) d x + 2 x y d y = 0 Rewrite the above equation in derivate form. ( x 2 + y 2 ) d x = − 2 x y d y d y d x = − ( x 2 + y 2 ) 2 x y Simplify the above equation. d y d x = − ( x 2 + y 2 ) 2 x y = − ( x 2 2 x y + y 2 2 x y ) = − 1 2 ( x y + y x ) … ( 1 ) As equation can be written in y x form, it is a homogeneous equation. Let y = v x and differentiate w.r.t x . d y d x = v + x d v d x Substitute v for y x and v + x d v d x for d y d x in equation ( 1 ) . v + x d v d x = − 1 2 ( 1 v + v ) x d v d x = − 1 2 ( 1 v + v ) − v x d v d x = − 1 2 v − v 2 − v x d v d x = − 1 − 3 v 2 2 v Separate the variable. 2 v − 1 − 3 v 2 d v = 1 x d x Integrate the above equation on both sides. ∫ 2 v − 1 − 3 v 2 d v = ∫ 1 x d x ∫ 2 v 1 + 3 v 2 d v = − ∫ 1 x d x 1 3 ∫ 6 v 1 + 3 v 2 d v = − ∫ 1 x d x … ( 2 ) Let 3 v 2 = t , then

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 2.6, Problem 12E

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The equation (x2+y2)dx+2xydy=0 Using Homogeneous equation method.

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Chapter 2.6, Problem 11E

Chapter 2.6, Problem 13E

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Fundamentals of Differential Equations and Boundary Value Problems

Ch. 2.2 - Prob. 1E Ch. 2.2 - In Problems 1-6, determine whether the given...Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E

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The equation ( x 2 + y 2 ) d x + 2 x y d y = 0 Using Homogeneous equation method.

To solve: The equation (x2+y2)dx+2xydy=0 Using Homogeneous equation method.

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Chapter 2 Solutions

To solve:

The equation (x2+y2)dx+2xydy=0 Using Homogeneous equation method.