https://file.aiquickdraw.com/mathbot/file/17284256581818p498oj6.jpg

Let's break down the solution provided in the image step by step for solving the given differential equation.

### Given Problem
The differential equation to be solved is:
\[
\frac{dy}{dx} + \frac{2x}{x^2 + 1}y = \frac{ux}{x^2 + 1}
\]
This is a first-order linear differential equation and the method used to solve it is through the "integrating factor" method.

---

### Step 1: Recognize the Standard Form

The equation can be rewritten in a standard linear differential equation form:

\[
\frac{dy}{dx} + P(x) y = Q(x)
\]
where \( P(x) = \frac{2x}{x^2 + 1} \) and \( Q(x) = \frac{ux}{x^2 + 1} \).

---

### Step 2: Identifying the Integrating Factor

The next step is to find the integrating factor \( \mu(x) \), which is given by:
\[
\mu(x) = e^{\int P(x) dx}
\]
Here, \( P(x) = \frac{2x}{x^2 + 1} \).

To calculate the integral, we perform:
\[
\int \frac{2x}{x^2 + 1} dx
\]
This is a standard integral, and its solution is:
\[
\int \frac{2x}{x^2 + 1} dx = \ln(x^2 + 1)
\]

Thus, the integrating factor is:
\[
\mu(x) = e^{\ln(x^2 + 1)} = x^2 + 1
\]

---

### Step 3: Multiply Through by the Integrating Factor

Next, we multiply the entire differential equation by the integrating factor \( \mu(x) = x^2 + 1 \):
\[
(x^2 + 1) \frac{dy}{dx} + \frac{2x}{x^2 + 1} (x^2 + 1) y = \frac{ux}{x^2 + 1} (x^2 + 1)
\]

This simplifies to:
\[
(x^2 + 1) \frac{dy}{dx} + 2x y = ux
\]

---

### Step 4: Recognize the Left-Hand Side as a Derivative

The left-hand side of the equation can now be recognized as the derivative of the product \( (x^2 + 1)y \). So the equation becomes:
\[
\frac{d}{dx} \left( (x^2 + 1) y \right) = ux
\]

---

### Step 5: Integrate Both Sides

To solve for \( y \), integrate both sides of the equation with respect to \( x \):

\[
\int \frac{d}{dx} \left( (x^2 + 1) y \right) dx = \int ux dx
\]

The left-hand side simplifies to:
\[
(x^2 + 1) y
\]

For the right-hand side, \( \int ux dx \) results in:
\[
\frac{u x^2}{2} + C
\]

Thus, the equation becomes:
\[
(x^2 + 1) y = \frac{u x^2}{2} + C
\]

---

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

Finally, solve for \( y \) by dividing both sides by \( (x^2 + 1) \):
\[
y = \frac{\frac{u x^2}{2} + C}{x^2 + 1}
\]

Or equivalently:
\[
y = \frac{u x^2}{2(x^2 + 1)} + \frac{C}{x^2 + 1}
\]

This is the general solution to the given differential equation.

---

### Summary of Steps:
1. Recognize the differential equation is linear and rewrite it in the standard form.
2. Identify \( P(x) \) and compute the integrating factor \( \mu(x) = e^{\int P(x) dx} \).
3. Multiply the entire equation by the integrating factor.
4. Recognize the left-hand side as the derivative of a product and rewrite.
5. Integrate both sides of the equation.
6. Solve for \( y \) to get the general solution.

---

Would you like more details on any of the steps, or have any questions?

Here are 5 follow-up questions for you:
1. How is the integrating factor \( \mu(x) \) used in linear differential equations?
2. Why does multiplying by the integrating factor help solve the differential equation?
3. Can you explain why the integral \( \int \frac{2x}{x^2 + 1} dx = \ln(x^2 + 1) \)?
4. What is the role of the arbitrary constant \( C \) in the solution?
5. How can this method be applied to other types of first-order differential equations?

**Tip:** Always check that the differential equation is in the standard form before finding the integrating factor. This ensures the solution process goes smoothly.

Solve the following differential equation: dy/dx + (2x)/(x^2 + 1)y = (ux)/(x^2 + 1).

This guide explains how to solve the first-order linear differential equation dy/dx + (2x)/(x^2 + 1)y = (ux)/(x^2 + 1) using the integrating factor method. The solution involves recognizing the standard form, finding the integrating factor, and solving step by step. Suitable for undergraduate students studying calculus or differential equations.

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