The given differential equation is:

$\frac{dy}{dx} + xy = x$

This is a first-order linear differential equation. To solve it, we follow these steps:

1. Identify the integrating factor:
   The equation can be rewritten in the standard form:
   $\frac{dy}{dx} + P(x)y = Q(x)$ Here, $P(x) = x$ and $Q(x) = x$. The integrating factor is: $\mu(x) = e^{\int P(x) dx} = e^{\int x dx} = e^{x^2/2}$

2. Multiply through by the integrating factor: $e^{x^2/2} \frac{dy}{dx} + e^{x^2/2} xy = e^{x^2/2} x$

3. Simplify the left side:
   The left side becomes the derivative of $(e^{x^2/2} y)$: $\frac{d}{dx}(e^{x^2/2} y) = e^{x^2/2} x$

4. Integrate both sides: $e^{x^2/2} y = \int e^{x^2/2} x dx$ Using substitution $u = \frac{x^2}{2}$, the integral simplifies to: $e^{x^2/2} y = e^{x^2/2} + C$

5. Solve for $y$: $y = 1 + C e^{-x^2/2}$

Thus, the solution is: $y = 1 + C e^{-x^2/2}$

Would you like more details or have any questions?

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Here are 5 related questions:

1. What is an integrating factor in differential equations?
2. How do you recognize linear differential equations?
3. Why is the substitution method used in integration?
4. Can this method be applied to non-linear differential equations?
5. How does the constant $C$ affect the general solution?

Tip: For first-order linear differential equations, always look for the integrating factor to simplify the solution process.