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verifique se a equação dada é exata. Se for, resolva. 

To verify if the given differential equation is exact and solve it if it is, let's start by analyzing the equation. The equation is:

\[
\left(\frac{3y^2 - x^2}{y^5}\right)\frac{dy}{dx} + \frac{x}{2y^4} = 0
\]

First, rewrite the equation in the standard form:

\[
M(x, y) + N(x, y) \frac{dy}{dx} = 0
\]

In this case:

\[
M(x, y) = \frac{x}{2y^4}, \quad N(x, y) = \frac{3y^2 - x^2}{y^5}
\]

### Step 1: Check if the equation is exact
To determine if the equation is exact, we need to check if the following condition holds:

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]

Let's calculate each partial derivative:

1. **Partial derivative of \( M(x, y) \) with respect to \( y \):**

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}\left(\frac{x}{2y^4}\right) = -\frac{2x}{2y^5} = -\frac{x}{y^5}
\]

2. **Partial derivative of \( N(x, y) \) with respect to \( x \):**

\[
\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}\left(\frac{3y^2 - x^2}{y^5}\right) = \frac{-2x}{y^5}
\]

### Step 2: Compare the partial derivatives
The partial derivatives are:

\[
\frac{\partial M}{\partial y} = -\frac{x}{y^5}, \quad \frac{\partial N}{\partial x} = \frac{-2x}{y^5}
\]

Since \(\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}\), the differential equation is **not exact**.

### Step 3: Solve the differential equation
Since the equation is not exact, we would typically need to find an integrating factor to make it exact. However, without further details or specific instructions, this step involves identifying an appropriate integrating factor that depends on either \(x\), \(y\), or both. 

Would you like to proceed with finding an integrating factor and solving the differential equation, or would you like additional details on this process?

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### Related Questions:

1. How do you find an integrating factor when the equation is not exact?
2. What is the general method for solving exact differential equations?
3. Can a differential equation be made exact by multiplying by a function of both \(x\) and \(y\)?
4. What are the conditions under which a differential equation can be considered exact?
5. How would the solution change if the initial condition \( y(1) = 1 \) were different?

### Tip:
When a differential equation is not exact, try looking for an integrating factor, which can often be a function of \(x\), \(y\), or both. This function, when multiplied with the original equation, can make it exact, allowing you to solve it using standard methods.

Learn how to verify if a differential equation is exact and solve it with a step-by-step guide. This problem involves differential equations, exact equations, and integrating factors. Suitable for advanced undergraduate students and anyone interested in differential equations.

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