The given differential equation is:

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

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

### Step 1: Write the equation in standard form
The equation is already in the standard form of a first-order linear differential equation:

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

Where \( P(x) = 2 \) and \( Q(x) = x \).

### 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 both sides of the equation by the integrating factor
Multiplying both sides of the differential equation by \( e^{2x} \):

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

The left side is the derivative of \( y e^{2x} \), so we can write:

\[
\frac{d}{dx} \left( y e^{2x} \right) = x e^{2x}
\]

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

\[
y e^{2x} = \int x e^{2x} \, dx
\]

We can solve the integral on the right-hand side using integration by parts. Let:

- \( u = x \), so \( du = dx \)
- \( dv = e^{2x} dx \), so \( v = \frac{1}{2} e^{2x} \)

Now apply the integration by parts formula:

\[
\int x e^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{2} \int e^{2x} dx
\]

\[
= \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + C
\]

### Step 5: Solve for \( y \)
Now substitute this back into the equation:

\[
y e^{2x} = \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + C
\]

Divide through by \( e^{2x} \):

\[
y = \frac{1}{2} x - \frac{1}{4} + C e^{-2x}
\]

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

\[
y(x) = \frac{1}{2} x - \frac{1}{4} + C e^{-2x}
\]

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

Would you like more details on any part of this solution?

### Related Questions:
1. How do you find the integrating factor for non-constant \( P(x) \)?
2. Can this method solve all first-order linear differential equations?
3. What is integration by parts and how does it work?
4. What is the physical interpretation of a first-order linear differential equation?
5. How does the general solution change if the equation has different initial conditions?

### Tip:
Always check the standard form of a differential equation before applying the integrating factor method—it simplifies solving significantly!

Learn how to solve the first-order linear differential equation dy/dx + 2y = x using the integrating factor method. This step-by-step guide covers essential concepts like finding the integrating factor and applying integration by parts. Suitable for Grades 11-12 and college-level students.

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