Explanation Given information: The provided differential equation is ( 2 x y 2 + 3 x 2 y − y 3 ) d x + ( 2 x 2 y + x 3 − 3 x y 2 ) d y = 0 . Concept used: The differential equation of the form, d y d x = f ( y x ) is said to be homogeneous. And we solve it by taking v = y x . The general solution is given by, d v f ( v ) − v = d x x . Consider the provided differential equation ( 2 x y 2 + 3 x 2 y − y 3 ) d x + ( 2 x 2 y + x 3 − 3 x y 2 ) d y = 0 . The given equation can also be written as, ( 2 x y 2 + 3 x 2 y − y 3 ) d x + ( 2 x 2 y + x 3 − 3 x y 2 ) d y = 0 ( 2 ( y 2 x 2 ) + 3 ( y x ) − ( y 3 x 3 ) ) d x + ( 2 ( y x ) + 1 − 3 ( y 2 x 2 ) ) d y = 0 d y d x = − ( 2 ( y 2 x 2 ) + 3 ( y x ) − ( y 3 x 3 ) ) ( 2 ( y x ) + 1 − 3 ( y 2 x 2 ) ) Now put v = y x ,then d y d x = − ( 2 ( v 2 ) + 3 ( v ) − ( v 3 ) ) ( 2 ( v ) + 1 − 3 ( v 2 ) ) = − 2 v 2 − 3 v + v 3 2 v + 1 − 3 v 2 Let f ( v ) = − 2 v 2 − 3 v + v 3 2 v + 1 − 3 v 2 , then we get &e

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 14.1, Problem 33PS

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Whether the equation (2xy2+3x2y−y3)dx+(2x2y+x3−3xy2)dy=0 is separable, homogenous, first-order linear, or exact, and then find the solution.

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Chapter 14.1, Problem 32PS

Chapter 14.1, Problem 34PS

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Whether the equation ( 2 x y 2 + 3 x 2 y − y 3 ) d x + ( 2 x 2 y + x 3 − 3 x y 2 ) d y = 0 is separable, homogenous, first-order linear, or exact, and then find the solution.

Solution Summary: The author explains that the differential equation (2xy2+3x

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Whether the equation (2xy2+3x2y−y3)dx+(2x2y+x3−3xy2)dy=0 is separable, homogenous, first-order linear, or exact, and then find the solution.

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