Explanation Given information: The given differential equation is [ 9 + ln ( x y 3 ) ] d x + ( 3 x y ) d y = 0 with initial condition y ( 1 ) = 1 . Concept Used: 1) Let M and N have continuous partial derivatives on an open disk R . The differential equation M ( x , y ) d x + N ( x , y ) d y = 0 is exact if and only if ∂ M ∂ y = ∂ N ∂ x 2) The equation M ( x , y ) d x + N ( x , y ) d y = 0 is an exact differential equation when there exists a function f of two variables x and y having continuous partial derivatives such that f x ( x , y ) = M ( x , y ) and f y ( x , y ) = N ( x , y ) The general solution of the equation is f y ( x , y ) = C Calculation: First we will compare differential equation [ 9 + ln ( x y 3 ) ] d x + ( 3 x y ) d y = 0 with the M ( x , y ) d x + N ( x , y ) d y = 0 . We have M = 9 + ln ( x y 3 ) N = 3 x y ∂ M ∂ y = ∂ [ 9 + ln ( x y 3 ) ] ∂ y ∂ M ∂ y = 3 y ∂ N ∂ x = ∂ ( 3 x y ) ∂ x ∂ N ∂ x = 3 y Since ∂ M ∂ y = ∂ N ∂ x , hence this differential equation is exact

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ISBN: 9781337604796

Author: Larson

Publisher: CENGAGE L

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Chapter 16, Problem 12RE

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The particular solution of the differential equation [9+ln(xy3)]dx+(3xy)dy=0 that satisfies the initial condition y(1)=1.

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The particular solution of the differential equation [ 9 + ln ( x y 3 ) ] d x + ( 3 x y ) d y = 0 that satisfies the initial condition y ( 1 ) = 1 .

Solution Summary: The author explains that the differential equation left[9+mathrmln (xy3) right]dx+ satisfies the initial condition

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The particular solution of the differential equation [9+ln(xy3)]dx+(3xy)dy=0 that satisfies the initial condition y(1)=1.

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