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.