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The problem asks you to analyze a differential equation and follow through a few steps. Let’s break down the problem and provide the solution:

### Given Differential Equation:

\[
\left( 9xe^{-\frac{1}{y^2}} + 16x^6 \right) + \left( 28x^{-1}y^5 + 9x^2 y e^{-\frac{1}{y^2}} \right) y' = 0
\]

### a) **Exactness Test**:
To check if the equation is exact, we need to compare the mixed partial derivatives of the functions \( M(x, y) \) and \( N(x, y) \), where:

- \( M(x, y) = 9xe^{-\frac{1}{y^2}} + 16x^6 \)
- \( N(x, y) = 28x^{-1}y^5 + 9x^2 y e^{-\frac{1}{y^2}} \)

Now, check the partial derivatives:

\[
M_y = \frac{\partial}{\partial y} \left( 9xe^{-\frac{1}{y^2}} + 16x^6 \right)
\]
\[
N_x = \frac{\partial}{\partial x} \left( 28x^{-1}y^5 + 9x^2 y e^{-\frac{1}{y^2}} \right)
\]

If \( M_y \neq N_x \), the equation is not exact.

### b) **Find an Integrating Factor \( \mu \)**:
To make the equation exact, we need to find an integrating factor \( \mu \). Assume \( \mu \) is a function of \( y \) only (as suggested). We will solve for \( \mu \) by ensuring that the new equation becomes exact after multiplying by \( \mu(y) \).

### c) **General Solution**:
Using the integrating factor \( \mu \), solve for the general solution \( \Psi(x, y) \).

---

I'll proceed with calculating the derivatives to verify exactness and find the integrating factor. Let me calculate this for you.The partial derivatives are:

\[
M_y = \frac{18x e^{-\frac{1}{y^2}}}{y^3}
\]
\[
N_x = 18x y e^{-\frac{1}{y^2}} - \frac{28y^5}{x^2}
\]

Clearly, \( M_y \neq N_x \), so the given equation is not exact.

### b) **Find the Integrating Factor \( \mu \)**:
We are tasked with finding an integrating factor \( \mu(y) \) that depends only on \( y \). To achieve this, I'll use the method to determine how \( \mu \) can make the equation exact.

Let me calculate this integrating factor.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, here is a guide you can follow to find the integrating factor \( \mu(y) \):

1. Using the formula for the integrating factor in terms of \( M \) and \( N \):
   \[
   \frac{\partial}{\partial y} \log(\mu(y)) = \frac{N_x - M_y}{M}
   \]
   where \( N_x - M_y \) is the difference between the partial derivatives we computed earlier, and \( M \) is the given expression for \( M(x, y) \).

2. Simplify the right-hand side and integrate with respect to \( y \) to find \( \mu(y) \).

After finding \( \mu(y) \), multiply the whole differential equation by \( \mu(y) \) and proceed to solve it.

If you would like more detailed help with manual steps or have any questions, feel free to ask!

Here are some related problems you could explore:

1. How do you determine the exactness of a differential equation?
2. What is an integrating factor and how is it used in solving differential equations?
3. Can an integrating factor depend on both \( x \) and \( y \)?
4. How do you solve a first-order linear differential equation using an integrating factor?
5. What are some alternative methods for solving non-exact differential equations?

**Tip**: When dealing with exact equations, always compare \( M_y \) and \( N_x \) first to quickly check exactness!

Consider the differential equation: (9xe^(-1/y^2) + 16x^6) + (28x^(-1)y^5 + 9x^2 y e^(-1/y^2)) y' = 0. a) This equation is not exact because M_y does not equal N_x. b) Find an integrating factor, μ, to make the equation exact. c) Using the integrating factor, find the general solution.

Learn how to solve a differential equation that is not exact by finding an integrating factor and determining the general solution. This problem explores key concepts such as the exactness of differential equations and the use of an integrating factor μ(y). Suitable for undergraduate students studying differential equations.

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