Explanation Approach: If right hand side of a differential equation can be expressed as a function of y x alone, then the differential equation is called homogenous differential equation. For solving homogenous differential equation the required substitution is y = v x . Calculation: Consider the given equation. ( x 2 + y 2 ) d x + 3 x y d y = 0 Write the above equation in the form of homogeneous differential equation as follows, d y d x = − x 2 − y 2 3 x y …… ( 1 ) Take y = v x Differentiate above expression with respect to x obtained d y d x as follows, d y d x = v + x d v d x Substitute v + x d v d x for d y d x and v x for y in equation ( 1 ) . v + x d v d x = − x 2 − ( v x ) 2 3 x ( v x ) = − x 2 − v 2 x 2 3 x 2 v = x 2 ( − 1 − v 2 ) 3 x 2 v = − 1 − v 2 3 v Further simplify, x d v d x = − 1 − v 2 3 v − v x d v d x = − 1 − v 2 − 3 v 2 3 v x d v d x = − 1 + 4 v 2 3 v Separate the variable for the above expression as follows, 3 v d v 1 + 4 v 2 = − d x x Let 1 + 4 v 2 = t Hence, 8 v d v = d t v d v = d t 8 Substitute t for 1 + 4 v 2 and d t 8 for v d v in above separated expression form as follows. 3 ( 1 t ) ( d t 8 ) = − d x x Integrate on both sides.

7th Edition

ISBN: 9780321977175

Author: Nagle

Publisher: PEARSON CO

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Chapter 2.RP, Problem 9RP

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The equation (x2+y2)dx+3xydy=0.

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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 + 3 x y d y = 0 .

To solve: The equation (x2+y2)dx+3xydy=0.

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Chapter 2 Solutions

To solve:

The equation (x2+y2)dx+3xydy=0.