### 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{\;}

Answered: Image /qna-images/answer/5bcae41f-a72b-4347-81b3-99dd7adea27c.jpg

Answered: If the new DE is written in the form…

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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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 differential equations.

Question

### 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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