The differential equation is:

\[
y' + y = 3x
\]

This is a first-order linear differential equation. We can solve it using an **integrating factor**. Here's the step-by-step process:

### Step 1: Write the equation in standard form

The standard form of a first-order linear differential equation is:

\[
y' + P(x) y = Q(x)
\]

In this case, we can identify:
- \( P(x) = 1 \)
- \( Q(x) = 3x \)

### Step 2: Find the integrating factor

The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int 1 \, dx} = e^x
\]

### Step 3: Multiply the equation by the integrating factor

Multiplying both sides of the differential equation by \( e^x \):

\[
e^x y' + e^x y = 3x e^x
\]

### Step 4: Recognize the left-hand side as a derivative

The left-hand side of the equation can now be written as the derivative of the product \( y e^x \):

\[
\frac{d}{dx} (y e^x) = 3x e^x
\]

### Step 5: Integrate both sides

Now, integrate both sides with respect to \( x \):

\[
y e^x = \int 3x e^x \, dx
\]

We can use **integration by parts** to solve the integral \( \int 3x e^x \, dx \). Let:
- \( u = x \), so \( du = dx \),
- \( dv = e^x dx \), so \( v = e^x \).

Now apply the integration by parts formula:

\[
\int u \, dv = uv - \int v \, du
\]

This gives:

\[
\int 3x e^x \, dx = 3 \left( x e^x - \int e^x \, dx \right) = 3(x e^x - e^x) = 3e^x(x - 1)
\]

Thus, we have:

\[
y e^x = 3e^x(x - 1) + C
\]

### Step 6: Solve for \( y \)

Divide both sides by \( e^x \):

\[
y = 3(x - 1) + C e^{-x}
\]

### Final solution:

The general solution to the differential equation is:

\[
y(x) = 3(x - 1) + C e^{-x}
\]

where \( C \) is an arbitrary constant determined by initial conditions if provided.

---

Do you have any questions or would you like further details?

### Related questions:
1. What are integrating factors, and how do they work in solving differential equations?
2. Can you show how to apply integration by parts more clearly?
3. What are some examples of initial conditions that could specify \( C \) here?
4. How can this method be extended to non-linear differential equations?
5. Are there any alternative methods to solve this type of equation?

### Tip: 
When solving first-order linear differential equations, always check if the equation is in standard form to apply the integrating factor method easily.

Learn how to solve the first-order linear differential equation y' + y = 3x using the integrating factor method. This detailed solution covers key mathematical concepts such as integrating factors, integration by parts, and the general solution of linear differential equations. Ideal for undergraduate calculus students.

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