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The given differential equation is:

\[
3e^{x} \tan y \, dx + \left(1 - e^{x}\right) \sec^2 y \, dy = 0
\]

### Steps to Solve the Differential Equation:

1. **Rearrange the Equation:**
   \[
   \frac{3e^{x} \tan y}{\left(1 - e^{x}\right) \sec^2 y} \, dx + dy = 0
   \]

   Simplify the expression inside the equation:
   \[
   \frac{3e^{x} \tan y \cos^2 y}{1 - e^{x}} \, dx + dy = 0
   \]

2. **Separate the Variables:**
   To separate variables, we need to express all \(x\) terms on one side and all \(y\) terms on the other side. Let’s rearrange:
   \[
   \frac{dy}{dx} = -\frac{3e^{x} \tan y \cos^2 y}{1 - e^{x}}
   \]
   Simplifying further:
   \[
   \frac{dy}{\tan y \cos^2 y} = -\frac{3e^{x}}{1 - e^{x}} \, dx
   \]

   We can rewrite \(\tan y \cos^2 y\) as \(\sin y \cos y\).

   Hence:
   \[
   \frac{dy}{\sin y \cos y} = -\frac{3e^{x}}{1 - e^{x}} \, dx
   \]

3. **Integrate Both Sides:**
   \[
   \int \frac{1}{\sin y \cos y} \, dy = -3 \int \frac{e^{x}}{1 - e^{x}} \, dx
   \]

   The integral on the left side:
   \[
   \int \frac{1}{\sin y \cos y} \, dy = \int \frac{2}{\sin 2y} \, dy = \int \csc 2y \, dy
   \]

   The integral on the right side requires substitution:
   \[
   u = 1 - e^{x}, \quad du = -e^{x} dx
   \]
   Therefore, the integral becomes:
   \[
   \int \frac{-3}{u} \, du = -3 \ln |u| = -3 \ln |1 - e^{x}|
   \]

   Thus:
   \[
   \ln |\csc 2y + \cot 2y| = 3 \ln |1 - e^{x}| + C
   \]

4. **Final Solution:**
   Simplifying, we have:
   \[
   \ln |\csc 2y + \cot 2y| = 3 \ln |1 - e^{x}| + C
   \]

   Therefore, the general solution is:
   \[
   |\csc 2y + \cot 2y| = C(1 - e^{x})^3
   \]

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

Would you like further details or have any questions? Here are some related questions for further practice:

1. How do you solve a linear differential equation?
2. What is the general solution of a first-order separable differential equation?
3. How can you apply substitution methods to simplify integrals in differential equations?
4. What is the significance of the constant of integration in a differential equation solution?
5. How can you verify if a given solution satisfies a differential equation?

**Tip:** When solving differential equations, always check if the equation can be simplified or separated into variables that can be integrated easily.

Solve the differential equation: 3e^x tan(y) dx + (1 - e^x) sec^2(y) dy = 0.

Step-by-step solution to solve the differential equation 3e^x tan(y) dx + (1 - e^x) sec^2(y) dy = 0 using separation of variables and integration techniques. Ideal for undergraduate students studying calculus.

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