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Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
BuyFindarrow_forward

Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16.1, Problem 8E
Textbook Problem
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Solving an Exact Differential Equation In Exercises 7–14, verify that the differential equation is exact. Then find the general solution.

y e x d x + e x d y = 0

To determine

Whether the differential equation yexdx+exdy=0 is exact or not and also finds the general solution.

Explanation of Solution

Given information:

The given differential equation is yexdx+exdy=0.

Concept Used:

1) Let M and N have continuous partial derivatives on an open disk R. The differential equation

M(x,y)dx+N(x,y)dy=0

is exact if and only if

My=Nx

2) The equation M(x,y)dx+N(x,y)dy=0 is an exact differential equation when there exists a function f of two variables x and y having continuous partial derivatives such that

fx(x,y)=M(x,y) and fy(x,y)=N(x,y)

The general solution of the equation is fy(x,y)=C

Calculation:

First we will compare differential equation yexdx+exdy=0 with the M(x,y)dx+N(x,y)dy=0.

We have

M=yexN=exMy=(yex)yMy=exNx=(ex)xNx=ex

Since My=Nx, hence this differential equation is exact

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Chapter 16 Solutions

Multivariable Calculus
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