Explanation Given information: The given differential equation is e − ( x 2 + y 2 ) ( x d x + y d y ) = 0 . 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 e − ( x 2 + y 2 ) ( x d x + y d y ) = 0 with the M ( x , y ) d x + N ( x , y ) d y = 0 . We have M = x e − ( x 2 + y 2 ) N = y e − ( x 2 + y 2 ) ∂ M ∂ y = ∂ ( x e − ( x 2 + y 2 ) ) ∂ y ∂ M ∂ y = − 2 x y ⋅ e − ( x 2 + y 2 ) ∂ N ∂ x = ∂ ( y e − ( x 2 + y 2 ) ) ∂ x ∂ N ∂ x = − 2 x y ⋅ e − ( x 2 + y 2 ) Since ∂ M ∂ y = ∂ N ∂ x , hence this differential equation is exact

11th Edition

ISBN: 9781337275378

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 16.1, Problem 12E

To determine

Whether the differential equation e−(x2+y2)(xdx+ydy)=0 is exact or not and also finds the general solution.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 16.1, Problem 11E

Chapter 16.1, Problem 13E

Students have asked these similar questions

MATH 1001(001-081) Question #9 Please answer the following. a) Populations of birds and insects are modeled by the differential equations shown below. Which of the variables, x or y, represents the birds and which represents the insects? dx = 0.4x-0.002xy dt dy = -0.2y+0.000008xy dt b) Please verify that y=4x-12x² +3 is a solution to the differential equation y"-2xy' +8y = 0.

solve the differential equation

Advanced Math

Chapter 16 Solutions

Multivariable Calculus

Ch. 16.1 - Exactness What does it mean for the...Ch. 16.1 - Integrating Factor When is it beneficial to use an...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Whether the differential equation e − ( x 2 + y 2 ) ( x d x + y d y ) = 0 is exact or not and also finds the general solution.

Solution Summary: The author compares the differential equation e-(x2+y 2)(xdx+ydy)=0 with the general solution.

Concept explainers

Videos

Whether the differential equation e−(x2+y2)(xdx+ydy)=0 is exact or not and also finds the general solution.

Want to see the full answer?

Chapter 16 Solutions