Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x, y) whose differential, d F(x, y) is the hand side of the differential equation. That is, level curves F(x, y) = C are solutions to the differential equation (2e* sin(y) – 3y)dx + (-3x + 2e* cos(y))dy = 0 First, if this equation has the form M(x, y)dx + N(x, y)dy = 0: M,(x, y) = , and N(x, y) = If the equation is not exact, enter not exact, otherwise enter in F(x, y) here

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
**Mixed Partials Check for Exact Differential Equations**

Use the "mixed partials" check to determine if the following differential equation is exact. If it is exact, identify a function \( F(x, y) \) whose differential, \( dF(x, y) \), matches the left-hand side of the differential equation. In other words, the level curves \( F(x, y) = C \) are solutions to the differential equation.

\[
(2e^x \sin(y) - 3y)dx + (-3x + 2e^x \cos(y))dy = 0
\]

Firstly, express the equation in the form \( M(x, y)dx + N(x, y)dy = 0 \):

- \( M_y(x, y) = \) **[Input Box]**

- \( N_x(x, y) = \) **[Input Box]**

If the equation is not exact, enter "not exact"; otherwise, enter the potential function \( F(x, y) \) here:

- **[Input Box]**
Transcribed Image Text:**Mixed Partials Check for Exact Differential Equations** Use the "mixed partials" check to determine if the following differential equation is exact. If it is exact, identify a function \( F(x, y) \) whose differential, \( dF(x, y) \), matches the left-hand side of the differential equation. In other words, the level curves \( F(x, y) = C \) are solutions to the differential equation. \[ (2e^x \sin(y) - 3y)dx + (-3x + 2e^x \cos(y))dy = 0 \] Firstly, express the equation in the form \( M(x, y)dx + N(x, y)dy = 0 \): - \( M_y(x, y) = \) **[Input Box]** - \( N_x(x, y) = \) **[Input Box]** If the equation is not exact, enter "not exact"; otherwise, enter the potential function \( F(x, y) \) here: - **[Input Box]**
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