To solve the first-order linear differential equation:

\[
\frac{dy}{dx} + 2y = x^3 - 7x^2 - 4x - 4
\]

We can use the **integrating factor method**. Here's the step-by-step process:

### Step 1: Rewrite the equation in standard form
The equation is already in standard form, which is:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

where \( P(x) = 2 \) and \( Q(x) = x^3 - 7x^2 - 4x - 4 \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) dx} = e^{\int 2 dx} = e^{2x}
\]

### Step 3: Multiply the equation by the integrating factor
Multiplying the entire equation by \( e^{2x} \) gives:

\[
e^{2x} \frac{dy}{dx} + 2e^{2x} y = e^{2x}(x^3 - 7x^2 - 4x - 4)
\]

The left-hand side becomes the derivative of \( y \cdot e^{2x} \):

\[
\frac{d}{dx}(y \cdot e^{2x}) = e^{2x}(x^3 - 7x^2 - 4x - 4)
\]

### Step 4: Integrate both sides
Now, integrate both sides with respect to \( x \):

\[
y \cdot e^{2x} = \int e^{2x}(x^3 - 7x^2 - 4x - 4) \, dx
\]

This integral can be solved using integration by parts or a combination of standard techniques. For brevity, we express the solution of the right-hand side as:

\[
y \cdot e^{2x} = \left( \frac{e^{2x}}{2}(x^3 - \frac{21}{2}x^2 - 2x - 2) \right) + C
\]

### Step 5: Solve for \( y \)
Finally, solve for \( y \) by dividing both sides by \( e^{2x} \):

\[
y = \frac{x^3}{2} - \frac{21x^2}{4} - x - 1 + Ce^{-2x}
\]

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

### Final Solution:
The solution to the differential equation is:

\[
y(x) = \frac{x^3}{2} - \frac{21x^2}{4} - x - 1 + Ce^{-2x}
\]

Let me know if you'd like more details or have any questions!

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Here are some related questions:
1. How do you find the general solution of non-homogeneous linear equations?
2. What is the role of integrating factors in solving first-order linear differential equations?
3. Can integrating factors always be used for any first-order linear equation?
4. How does the integrating factor method compare with separation of variables?
5. How would you solve a second-order linear differential equation?

**Tip:** When solving a differential equation, always ensure it's in standard form before applying methods like integrating factors!

Explore how to solve the first-order linear differential equation dy/dx + 2y = x^3 - 7x^2 - 4x - 4 using the integrating factor method. This solution includes detailed steps on finding the integrating factor, multiplying the equation, and integrating both sides. Ideal for undergraduate students learning calculus and differential equations.

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