The given differential equation is:

\[
\sin(9x) \, dx + 2y \cos^3(9x) \, dy = 0
\]

Let's solve this step by step.

### Step 1: Rewrite the equation in separable form
We aim to separate variables \( x \) and \( y \). Start by isolating the terms involving \( dx \) and \( dy \):

\[
\sin(9x) \, dx = -2y \cos^3(9x) \, dy
\]

Now, divide both sides by \( 2y \cos^3(9x) \) to separate the variables:

\[
\frac{\sin(9x)}{\cos^3(9x)} \, dx = -2 \frac{dy}{y}
\]

### Step 2: Simplify the trigonometric expression
We can simplify \( \frac{\sin(9x)}{\cos^3(9x)} \) using the identity \( \tan(9x) = \frac{\sin(9x)}{\cos(9x)} \):

\[
\frac{\sin(9x)}{\cos^3(9x)} = \frac{\tan(9x)}{\cos^2(9x)}
\]

Thus, the equation becomes:

\[
\frac{\tan(9x)}{\cos^2(9x)} \, dx = -2 \frac{dy}{y}
\]

### Step 3: Integrate both sides
Now, we integrate both sides to find the solution.

- For the left side: 
\[
\int \frac{\tan(9x)}{\cos^2(9x)} \, dx
\]

This integral requires substitution. Let \( u = \cos(9x) \), so that \( du = -9 \sin(9x) dx \). The integral simplifies into terms of \( u \).

- For the right side:
\[
\int -2 \frac{dy}{y} = -2 \ln |y|
\]

I'll proceed with solving these integrals next.The integrals yield the following results:

- Left side: \(\int \frac{\tan(9x)}{\cos^2(9x)} \, dx = \frac{1}{18 \cos^2(9x)}\)
- Right side: \(\int -2 \frac{dy}{y} = -2 \ln |y|\)

### Step 4: Combine the results
Now, combining both integrals:

\[
\frac{1}{18 \cos^2(9x)} = -2 \ln |y| + C
\]

where \( C \) is the constant of integration.

This is the implicit solution to the differential equation.

Would you like to explore more details about any specific part of the solution?

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**Related Questions:**
1. How can we solve trigonometric integrals in general?
2. What substitution method can we use for similar trigonometric integrals?
3. How does separation of variables work in solving differential equations?
4. What are the properties of \(\tan(x)\) and \(\cos(x)\) in integration?
5. Can the solution be simplified further or written in a different form?

**Tip:** For differential equations, try applying separation of variables early to simplify the solving process.

Learn how to solve the trigonometric differential equation sin(9x) dx + 2y cos^3(9x) dy = 0 using separation of variables and trigonometric substitution. This detailed step-by-step solution covers essential calculus and trigonometric concepts. Ideal for undergraduate students.

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