We have the differential equation:

\[
y' = x - 2y \cot(x)
\]

This is a first-order linear differential equation. To solve it, we will use the integrating factor method. Let's rewrite the equation in standard linear form:

\[
y' + 2y \cot(x) = x
\]

### Step 1: Identify the integrating factor
For an equation in the form:

\[
y' + P(x)y = Q(x)
\]

The integrating factor is given by:

\[
\mu(x) = e^{\int P(x) \, dx}
\]

Here, \( P(x) = 2 \cot(x) \). So, we need to compute:

\[
\int 2 \cot(x) \, dx
\]

We know that:

\[
\int \cot(x) \, dx = \ln|\sin(x)|
\]

Thus:

\[
\int 2 \cot(x) \, dx = 2 \ln|\sin(x)| = \ln|\sin^2(x)|
\]

So, the integrating factor is:

\[
\mu(x) = e^{\ln|\sin^2(x)|} = \sin^2(x)
\]

### Step 2: Multiply through by the integrating factor
We multiply both sides of the equation by \( \sin^2(x) \):

\[
\sin^2(x) y' + 2y \sin(x) \cos(x) = x \sin^2(x)
\]

The left-hand side can be recognized as the derivative of \( y \sin^2(x) \):

\[
\frac{d}{dx}(y \sin^2(x)) = x \sin^2(x)
\]

### Step 3: Integrate both sides
Now integrate both sides with respect to \( x \):

\[
y \sin^2(x) = \int x \sin^2(x) \, dx
\]

The integral on the right side can be solved using integration by parts or a trigonometric identity. Let me calculate that next.from sympy import symbols, sin, integrate

# Define the variable and the function
x = symbols('x')
integrand = x * sin(x)**2

# Compute the integral of x * sin^2(x)
integral_result = integrate(integrand, x)
integral_resultThe integral of \( x \sin^2(x) \) is:

\[
\int x \sin^2(x) \, dx = \frac{x^2 \sin^2(x)}{4} + \frac{x^2 \cos^2(x)}{4} - \frac{x \sin(x) \cos(x)}{2} + \frac{\sin^2(x)}{4}
\]

### Step 4: Solve for \( y \)
Thus, the equation becomes:

\[
y \sin^2(x) = \frac{x^2 \sin^2(x)}{4} + \frac{x^2 \cos^2(x)}{4} - \frac{x \sin(x) \cos(x)}{2} + \frac{\sin^2(x)}{4} + C
\]

Now, divide through by \( \sin^2(x) \) to solve for \( y \):

\[
y = \frac{x^2}{4} + \frac{x^2 \cot^2(x)}{4} - \frac{x \cot(x)}{2} + \frac{1}{4} + \frac{C}{\sin^2(x)}
\]

This is the general solution of the given differential equation.

Would you like more details or clarification on any step?

Here are 5 related questions you might find useful:
1. How does the integrating factor method work for other first-order linear equations?
2. What is the role of trigonometric identities in solving differential equations?
3. Can the solution method change if the equation was non-linear?
4. How can integration by parts be used to solve integrals like \( \int x \sin^2(x) \, dx \)?
5. What are some typical initial conditions to apply for finding a particular solution?

**Tip:** When dealing with trigonometric functions, try simplifying the integrals using identities like \( \sin^2(x) + \cos^2(x) = 1 \).

Learn how to solve the differential equation y' = x - 2y cot(x) using the integrating factor method. Detailed step-by-step solution includes calculating the integrating factor, integrating both sides, and solving for the general solution. Suitable for advanced college-level mathematics.

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