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Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 1PS
Textbook Problem
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Finding a General Solution Find the value of k that makes the differential equation

( 3 x 2 + k x y 2 ) d x ( 5 x 2 y + k y 2 ) d y = 0

exact. Using this value of k, find the general solution.

To determine

To calculate: The value of k such that the differential equation (3x2+kxy2)dx(5x2y+ky2)dy=0 is exact. Also, use this value of k to find the general solution.

Explanation of Solution

Given information: The differential equation (3x2+kxy2)dx(5x2y+ky2)dy=0.

Formula used: 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.

Calculation:

Compare the differential equation (3x2+kxy2)dx(5x2y+ky2)dy=0 with the general form M(x,y)dx+N(x,y)dy=0 to get M(x,y)=(3x2+kxy2)andN(x,y)=(5x2y+ky2).

Now,

M(x,y)=(3x2+kxy2)My=2kxy

Also,

N(x,y)=(5x2y+ky2)Nx=10xy

As the given equationis exact My=Nx.

Thus,

2kxy=10xy

Comparing the coefficients of xy on both the sides

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Chapter 16 Solutions

Multivariable Calculus
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Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Graphical and Analytic Analysis In Exercises 15...Ch. 16.1 - Graphical and Analytic AnalysisIn Exercises 15 and...Ch. 16.1 - Finding a Particular Solution In Exercises17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Integrating Factor Show that each expression is...Ch. 16.1 - Integrating FactorShow that thedifferential...Ch. 16.1 - Tangent Curves In Exercises 39-42, use agraphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Finding an Equation of a Curve In Exercise 43 and...Ch. 16.1 - Finding an Equation of a Curve In Exercises 43 and...Ch. 16.1 - Cost In a manufacturing process where y=C(x)...Ch. 16.1 - HOW DO YOU SEE? 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