If the new DE is written in the form M(x, y) dx + N(x, y) dy = 0, one has My = || N, = X, Since M, and N, ---Select--- v equal, the equation is exact.
If the new DE is written in the form M(x, y) dx + N(x, y) dy = 0, one has My = || N, = X, Since M, and N, ---Select--- v equal, the equation is exact.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Verification and Solution of a Differential Equation
#### Verify that the given differential equation is not exact.
The given differential equation is:
\[ (-xy \sin(x) + 2y \cos(x)) \, dx + 2x \cos(x) \, dy = 0. \]
To verify if the given DE is written in the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), one has:
\[ M = -xy \sin(x) + 2y \cos(x) \]
\[ N = 2x \cos(x) \]
Next, compute the partial derivatives \( M_y \) and \( N_x \):
\[ M_y = -x \sin(x) + 2 \cos(x) \]
\[ N_x = 2 \cos(x) - 2x \sin(x) \]
Since \( M_y \) and \( N_x \) are not equal, the equation is not exact.
#### Multiply the given differential equation by the integrating factor \( \mu(x, y) = xy \) and verify that the new equation is exact.
Multiplying by the given integrating factor \( \mu(x, y) = xy \):
If the new DE is written in the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), one has:
\[ M_y = \]
\[ N_x = \]
Since \( M_y \) and \( N_x \) are equal, the equation is exact.
#### Solve.
\[\boxed{\;}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F19331858-c67c-414f-8328-5459040eae8a%2F5bcae41f-a72b-4347-81b3-99dd7adea27c%2Fhxxty4.png&w=3840&q=75)
Transcribed Image Text:### Verification and Solution of a Differential Equation
#### Verify that the given differential equation is not exact.
The given differential equation is:
\[ (-xy \sin(x) + 2y \cos(x)) \, dx + 2x \cos(x) \, dy = 0. \]
To verify if the given DE is written in the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), one has:
\[ M = -xy \sin(x) + 2y \cos(x) \]
\[ N = 2x \cos(x) \]
Next, compute the partial derivatives \( M_y \) and \( N_x \):
\[ M_y = -x \sin(x) + 2 \cos(x) \]
\[ N_x = 2 \cos(x) - 2x \sin(x) \]
Since \( M_y \) and \( N_x \) are not equal, the equation is not exact.
#### Multiply the given differential equation by the integrating factor \( \mu(x, y) = xy \) and verify that the new equation is exact.
Multiplying by the given integrating factor \( \mu(x, y) = xy \):
If the new DE is written in the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), one has:
\[ M_y = \]
\[ N_x = \]
Since \( M_y \) and \( N_x \) are equal, the equation is exact.
#### Solve.
\[\boxed{\;}
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