https://file.aiquickdraw.com/mathbot/file/1725722406048q9fquxzi.jpg

The image you provided contains a solution to a problem involving a differential equation. I'll break down the steps shown in the image in a simple way:

### Problem Statement
It seems the problem is about solving a differential equation of the form:

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

Where \( P(x) \) and \( Q(x) \) are functions of \( x \).

### Solution Overview
1. **Identify the Integrating Factor (IF):**
   - The integrating factor (IF) is found using the formula:
     \[
     \text{IF} = e^{\int P(x) \, dx}
     \]
   - In the image, it seems \( P(x) = \frac{2}{x} \), so the integrating factor is:
     \[
     \text{IF} = e^{\int \frac{2}{x} \, dx} = e^{2 \ln x} = x^2
     \]

2. **Multiply the Equation by the Integrating Factor:**
   - Multiply the entire differential equation by the integrating factor \( x^2 \):
     \[
     x^2 \frac{dy}{dx} + x^2 \cdot \frac{2}{x} y = x^2 \cdot Q(x)
     \]
   - Simplifying, the equation becomes:
     \[
     \frac{d}{dx} (x^2 y) = x^2 Q(x)
     \]
   - This simplifies because the left-hand side is the derivative of the product \( x^2 y \).

3. **Integrate Both Sides:**
   - Integrate both sides with respect to \( x \) to solve for \( y \):
     \[
     x^2 y = \int x^2 Q(x) \, dx + C
     \]
   - Here, \( C \) is the constant of integration.

4. **Solve for \( y \):**
   - Finally, divide by \( x^2 \) to isolate \( y \):
     \[
     y = \frac{1}{x^2} \left(\int x^2 Q(x) \, dx + C \right)
     \]

### Summary of Steps
1. Find the integrating factor.
2. Multiply the equation by the integrating factor.
3. Simplify to a form where the left side is the derivative of a product.
4. Integrate both sides.
5. Solve for \( y \).

Would you like to go into any more details or have any questions about the process? Here are some related questions:

1. What is an integrating factor, and why is it used in solving differential equations?
2. How do you compute the integral of a logarithmic function?
3. Can this method be applied to any first-order linear differential equation?
4. What are some common mistakes to avoid when using the integrating factor method?
5. How do you interpret the constant of integration in the solution?

**Tip:** When solving differential equations, always check if the equation is in a form that allows you to use known methods like the integrating factor technique.

Solve the differential equation \( \frac{dy}{dx} + \frac{2}{x} y = Q(x) \).

This solution walks you through solving the differential equation \( \frac{dy}{dx} + \frac{2}{x} y = Q(x) \) using the integrating factor method. Learn the step-by-step process of finding the integrating factor, multiplying both sides, and solving for y.

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