6𝑥𝑦3 + 2𝑦4 + (9𝑥2𝑦2 + 8𝑥𝑦3)𝑦′ = 0

To solve the given differential equation:

\[
6xy^3 + 2y^4 + \left(9x^2y^2 + 8xy^3\right)y' = 0
\]

we proceed with the following steps:

### Step 1: Identify the terms and structure
This is a first-order differential equation in the form:

\[
A(x, y) + B(x, y) y' = 0
\]

Where:
- \( A(x, y) = 6xy^3 + 2y^4 \)
- \( B(x, y) = 9x^2y^2 + 8xy^3 \)

### Step 2: Isolate the derivative term
To solve for \( y' \), isolate it on one side of the equation:

\[
\left(9x^2y^2 + 8xy^3\right)y' = -\left(6xy^3 + 2y^4\right)
\]

Now, solve for \( y' \):

\[
y' = \frac{-\left(6xy^3 + 2y^4\right)}{9x^2y^2 + 8xy^3}
\]

### Step 3: Simplify the expression
Simplify the expression for \( y' \):

\[
y' = \frac{-2y^3(3x + y)}{y^2(9x^2 + 8xy)}
\]

Further simplification gives:

\[
y' = \frac{-2y(3x + y)}{9x^2 + 8xy}
\]

This is the simplified form of the derivative \( y' \).

Would you like more details or have any questions?

Here are some related questions:
1. What are the different methods to solve first-order differential equations?
2. How do you determine if a differential equation is separable?
3. What are exact differential equations, and how are they solved?
4. Can you always simplify the expression for \( y' \) in any differential equation?
5. How can you verify the solution to a differential equation?

**Tip:** Always check if a differential equation can be simplified or transformed into a standard form before solving it.

Learn how to solve the first-order differential equation 6xy^3 + 2y^4 + (9x^2y^2 + 8xy^3)y' = 0 step-by-step. This problem involves identifying terms, isolating the derivative term, and simplifying the expression. Suitable for college-level mathematics.

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