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The given differential equation is:

\[
\frac{dy}{dx} + 3x^2 y = 6x^2
\]

This is a first-order linear differential equation, and we can solve it using an **integrating factor**.

### Step-by-step solution:

1. **Rewrite the equation in standard form**:
   \[
   \frac{dy}{dx} + P(x) y = Q(x)
   \]
   where \( P(x) = 3x^2 \) and \( Q(x) = 6x^2 \).

2. **Find the integrating factor**:
   The integrating factor \( \mu(x) \) is given by:
   \[
   \mu(x) = e^{\int P(x) \, dx} = e^{\int 3x^2 \, dx}
   \]
   \[
   \mu(x) = e^{x^3}
   \]

3. **Multiply the differential equation by the integrating factor**:
   Multiply both sides of the equation by \( e^{x^3} \):
   \[
   e^{x^3} \frac{dy}{dx} + e^{x^3} 3x^2 y = e^{x^3} 6x^2
   \]

4. **Simplify the left-hand side**:
   Notice that the left-hand side is the derivative of \( y e^{x^3} \) with respect to \( x \):
   \[
   \frac{d}{dx} \left( y e^{x^3} \right) = 6x^2 e^{x^3}
   \]

5. **Integrate both sides**:
   Now, integrate both sides with respect to \( x \):
   \[
   y e^{x^3} = \int 6x^2 e^{x^3} \, dx
   \]

   Let \( u = x^3 \), so \( du = 3x^2 \, dx \). Therefore:
   \[
   \int 6x^2 e^{x^3} \, dx = 2e^{x^3} + C
   \]

6. **Solve for \( y \)**:
   Divide both sides by \( e^{x^3} \):
   \[
   y = 2 + C e^{-x^3}
   \]

### Final solution:
\[
y(x) = 2 + C e^{-x^3}
\]

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

---

Would you like any further clarification or details on this solution?

Here are 5 related questions for further practice:
1. How would you solve the differential equation \( \frac{dy}{dx} + x y = x^2 \)?
2. What are the properties of integrating factors in solving linear differential equations?
3. How can we verify if a given function is a solution to a differential equation?
4. How do you solve homogeneous first-order differential equations?
5. How can you apply integrating factors to non-homogeneous linear equations?

**Tip:** When solving linear differential equations, always look for ways to simplify the equation through multiplication by an integrating factor.

Solve the differential equation \( \frac{dy}{dx} + 3x^2 y = 6x^2 \).

Learn how to solve the first-order linear differential equation \( \frac{dy}{dx} + 3x^2 y = 6x^2 \) using the integrating factor method. This detailed solution covers key concepts such as the standard form of a linear equation, finding the integrating factor, and solving for y. Suitable for college-level mathematics students.

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