Find the general solution of 𝑦
′ + 2𝑥𝑦 = 𝑥.

To find the general solution of the differential equation \( y' + 2xy = x \), we can use the method of integrating factors.

1. **Rewrite the equation in standard form:**
\[ y' + 2xy = x \]

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

3. **Multiply the entire differential equation by the integrating factor:**
\[ e^{x^2} y' + 2x e^{x^2} y = x e^{x^2} \]

4. **Rewrite the left-hand side as the derivative of a product:**
\[ \frac{d}{dx} \left( e^{x^2} y \right) = x e^{x^2} \]

5. **Integrate both sides with respect to \( x \):**
\[ \int \frac{d}{dx} \left( e^{x^2} y \right) \, dx = \int x e^{x^2} \, dx \]

The left-hand side integrates to:
\[ e^{x^2} y \]

For the right-hand side, let \(

Discover the general solution of the differential equation y' + 2xy = x using integrating factors and integration techniques. This detailed guide covers step-by-step methods to solve differential equations, focusing on integrating factors and their application. Suitable for undergraduate-level mathematics students.

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