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giải bài tập này theo chương trình toán 2 bậc đại học

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.

Tìm nghiệm riêng của phương trình 2xy + (x^2 + 1) dy/dx = 1; y(1) = 1

This detailed guide solves the differential equation 2xy + (x^2 + 1) dy/dx = 1 with the initial condition y(1) = 1. Learn the method of integrating factors to find the particular solution, ideal for students at the university level studying Calculus II or Differential Equations.

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