The given problem is asking to find the particular solution of the differential equation:

$2xy + (x^2 + 1) \frac{dy}{dx} = 1$

with the initial condition $y(1) = 1$.

Step 1: Rewrite the equation

We start by isolating $\frac{dy}{dx}$:

$(x^2 + 1) \frac{dy}{dx} = 1 - 2xy$

Thus, the equation becomes:

$\frac{dy}{dx} = \frac{1 - 2xy}{x^2 + 1}$

Step 2: Solve using the method of separable variables (if possible)

This is not directly separable, so we will use the method of integrating factor or another suitable method. Let's explore the form of the equation.

Step 3: Solve the first-order linear differential equation

This equation can be treated as a first-order linear differential equation of the form:

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

Rewriting the equation:

$\frac{dy}{dx} + \frac{2x}{x^2 + 1} y = \frac{1}{x^2 + 1}$

Here, $P(x) = \frac{2x}{x^2 + 1}$ and $Q(x) = \frac{1}{x^2 + 1}$.

Step 4: Find the integrating factor

The integrating factor $\mu(x)$ is given by:

$\mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{2x}{x^2 + 1} \, dx}$

The integral of $\frac{2x}{x^2 + 1}$ is $\ln(x^2 + 1)$, so the integrating factor is:

$\mu(x) = e^{\ln(x^2 + 1)} = x^2 + 1$

Step 5: Multiply the differential equation by the integrating factor

Multiply the entire differential equation by $x^2 + 1$:

$(x^2 + 1) \frac{dy}{dx} + 2x y = 1$

This simplifies to:

$\frac{d}{dx} \left( (x^2 + 1)y \right) = 1$

Step 6: Integrate both sides

Now, integrate both sides with respect to $x$:

$(x^2 + 1) y = \int 1 \, dx = x + C$

Thus, the general solution is:

$y = \frac{x + C}{x^2 + 1}$

Step 7: Apply the initial condition

Using the initial condition $y(1) = 1$, substitute $x = 1$ and $y = 1$:

$1 = \frac{1 + C}{1^2 + 1} = \frac{1 + C}{2}$

Solving for $C$:

$1 = \frac{1 + C}{2} \implies 2 = 1 + C \implies C = 1$

Final Solution:

The particular solution to the differential equation is:

$y = \frac{x + 1}{x^2 + 1}$

Would you like a further breakdown of any part of the solution, or do you have any questions?

Related Questions:

1. How do you determine whether a differential equation is separable or linear?
2. What is an integrating factor, and how is it used to solve differential equations?
3. Can all first-order linear differential equations be solved using integrating factors?
4. How would the solution change if the initial condition were different?
5. How does the method of variation of parameters compare to integrating factors?

Tip:

When solving first-order linear differential equations, always check if the equation can be simplified or transformed using integrating factors before trying more advanced methods.