https://file.aiquickdraw.com/mathbot/file/172935531739357l0sn0f.jpg

The given equation in the image is:

\[
(y - x y^2) dx - (x + x^2 y) dy = 0
\]

This is a first-order differential equation that can be solved by several methods, such as finding an integrating factor or checking if it is exact.

### Let's check if the equation is exact:
For an equation of the form:
\[
M(x, y) dx + N(x, y) dy = 0,
\]
the condition for exactness is:
\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.
\]
Here:
- \( M(x, y) = y - x y^2 \)
- \( N(x, y) = -(x + x^2 y) \).

Now, let's compute the partial derivatives:

1. \( \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(y - x y^2) = 1 - 2x y \).
2. \( \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(-(x + x^2 y)) = -(1 + 2x y) \).

Clearly, these partial derivatives are equal:
\[
1 - 2xy = -(1 + 2xy),
\]
which means the equation is **not exact**.

### Next step:
Since the equation is not exact, an integrating factor might be needed. Would you like to proceed with finding an integrating factor or explore another method of solving this differential equation?

Let me know if you'd like details on the process, and here are some further questions to expand your understanding:

1. What is an exact differential equation?
2. How do we find integrating factors for non-exact equations?
3. What are common methods to solve first-order differential equations?
4. What is the role of partial derivatives in determining exactness?
5. How can you verify if a given solution satisfies a differential equation?

**Tip:** Checking for exactness is a quick way to simplify solving many differential equations, as it allows the direct integration of the components.

Learn how to solve the first-order differential equation (y - xy^2) dx - (x + x^2 y) dy = 0 using methods such as exactness and integrating factors. This problem explores key concepts like partial derivatives and exact differential equations. Suitable for undergraduate students.

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