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Let's solve the two questions step by step.

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### Question 7:
We are asked to apply the method of **separation of variables** to the following differential equation:
\[
\csc(y) \, dx + \sec^2(x) \, dy = 0
\]

**Step 1: Rewrite the equation to separate variables:**

We want to separate \(x\) terms with \(dx\) and \(y\) terms with \(dy\). To do that, move terms with \(dx\) and \(dy\) to opposite sides:
\[
\csc(y) \, dx = -\sec^2(x) \, dy
\]

Now, divide both sides by \(\sec^2(x)\) and \(\csc(y)\) to fully separate the variables:
\[
\frac{dx}{\sec^2(x)} = - \frac{dy}{\csc(y)}
\]

**Step 2: Simplify each side:**

Use the identities \(\sec^2(x) = 1/\cos^2(x)\) and \(\csc(y) = 1/\sin(y)\):
\[
\cos^2(x) \, dx = - \sin(y) \, dy
\]

**Step 3: Integrate both sides:**

Now, integrate both sides:
\[
\int \cos^2(x) \, dx = - \int \sin(y) \, dy
\]

We use a standard trigonometric identity for the integral of \(\cos^2(x)\):
\[
\int \cos^2(x) \, dx = \frac{x}{2} + \frac{\sin(2x)}{4} + C_1
\]
For the right side:
\[
- \int \sin(y) \, dy = \cos(y) + C_2
\]

**Step 4: Combine the constants:**

We can combine the constants \(C_1\) and \(C_2\) into a single constant:
\[
\frac{x}{2} + \frac{\sin(2x)}{4} = \cos(y) + C
\]

This is the general solution after applying the method of separation of variables.

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### Question 8:
We are asked to solve the following differential equation:
\[
\sin(3x) \, dx + 2y \, \cos^3(3x) \, dy = 0, \quad y(\pi) = 0
\]

**Step 1: Separate variables:**

We rewrite the equation to separate \(x\)-terms and \(y\)-terms:
\[
\sin(3x) \, dx = -2y \, \cos^3(3x) \, dy
\]

Divide both sides by \(y\) and \(\sin(3x)\):
\[
\frac{dx}{\cos^3(3x)} = - \frac{2 \, dy}{y}
\]

**Step 2: Simplify and integrate:**

We can simplify and integrate both sides:
- The left side: Let \(u = 3x\), so \(du = 3 \, dx\):
  \[
  \int \frac{dx}{\cos^3(3x)} = \frac{1}{3} \int \sec^3(u) \, du
  \]
  The integral of \(\sec^3(u)\) is a standard result:
  \[
  \int \sec^3(u) \, du = \frac{1}{2} \sec(u) \tan(u) + \frac{1}{2} \ln|\sec(u) + \tan(u)| + C
  \]
  Substituting \(u = 3x\) back:
  \[
  \frac{1}{3} \left( \frac{1}{2} \sec(3x) \tan(3x) + \frac{1}{2} \ln|\sec(3x) + \tan(3x)| \right)
  \]

- The right side:
  \[
  \int -\frac{2 \, dy}{y} = -2 \ln|y| + C
  \]

**Step 3: Combine the results:**

Equating both sides:
\[
\frac{1}{6} \sec(3x) \tan(3x) + \frac{1}{6} \ln|\sec(3x) + \tan(3x)| = -2 \ln|y| + C
\]

**Step 4: Apply initial condition \(y(\pi) = 0\):**

Using the initial condition \(y(\pi) = 0\), we substitute \(x = \pi\) and \(y = 0\) into the equation and solve for the constant \(C\).

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Would you like more detailed steps, or do you have any further questions? Here are five related questions to explore further:

1. How do you solve a first-order linear differential equation?
2. What is the significance of boundary conditions in differential equations?
3. Can you derive the integral of \(\sec^3(x)\) from scratch?
4. How can you apply separation of variables to partial differential equations?
5. What are some common techniques for solving trigonometric integrals?

**Tip:** When solving differential equations, always simplify trigonometric identities early to make the integration steps easier!

What is the result of applying the variable separation method in csc(y) dx + sec^2(x) dy = 0? Find the solution to sin(3x) dx + 2y cos^3(3x) dy = 0, y(π) = 0.

Explore how to solve two differential equations using the separation of variables method. The problem involves csc(y) dx + sec^2(x) dy = 0 and sin(3x) dx + 2y cos^3(3x) dy = 0 with y(π) = 0, incorporating trigonometric identities and advanced calculus techniques.

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