Solve the differential equation dy/dx + y*cot(x) = 2*x + $x^2*cot(x)$ given y(pi/2) = 0.

Step 1: Identify the form of the differential equation

The given differential equation is of the form $\frac{dy}{dx} + y\cot(x) = 2x + x^2\cot(x)$. This is a first-order linear differential equation, which can be written in the standard form $\frac{dy}{dx} + P(x)y = Q(x)$. By comparing the given equation with the standard form, we can identify $P(x)$ and $Q(x)$.

Step 2: Identify P(x) and Q(x)

From the standard form, we can see that $P(x)$ is the coefficient of $y$ and $Q(x)$ is the term on the right-hand side of the equation. In this case, $P(x) = \cot(x)$ and $Q(x) = 2x + x^2\cot(x)$.

Step 3: Calculate the integrating factor

The integrating factor (IF) for a linear differential equation is given by the formula $e^{\int P(x)dx}$. We substitute $P(x) = \cot(x)$ into the formula and integrate. The integral of $\cot(x)$ is $\ln|\sin(x)|$. Assuming $\sin(x) > 0$ for its domain, the integrating factor simplifies to $\sin(x)$.

Step 4: Multiply the differential equation by the integrating factor

Now we multiply both sides of the original differential equation by the integrating factor, $\sin(x)$. This step transforms the left-hand side of the equation into the derivative of the product of $y$ and the integrating factor, which is $\frac{d}{dx}(y \cdot IF)$.

Step 5: Simplify the equation

The left-hand side of the equation becomes $\frac{d}{dx}(y\sin(x))$ because $\frac{d}{dx}(y\sin(x)) = \frac{dy}{dx}\sin(x) + y\cos(x)$. Since $\cot(x)\sin(x) = \frac{\cos(x)}{\sin(x)}\sin(x) = \cos(x)$, the equation simplifies to $\frac{d}{dx}(y\sin(x)) = 2x\sin(x) + x^2\cos(x)$. We can recognize the right-hand side as the derivative of $x^2\sin(x)$ using the product rule: $\frac{d}{dx}(x^2\sin(x)) = 2x\sin(x) + x^2\cos(x)$.

Step 6: Integrate both sides

We integrate both sides of the simplified equation with respect to $x$. The integral of $\frac{d}{dx}(y\sin(x))$ is simply $y\sin(x)$. On the right-hand side, we recognize $(2x\sin(x) + x^2\cos(x))$ as the derivative of $x^2\sin(x)$, so its integral is $x^2\sin(x) + C$, where $C$ is the constant of integration.

Step 7: Solve for y

To find the general solution for $y$, we divide both sides of the equation by $\sin(x)$. This gives us $y = \frac{x^2\sin(x) + C}{\sin(x)}$, which simplifies to $y = x^2 + C\csc(x)$.

Step 8: Apply the initial condition

We are given the initial condition $y(\frac{\pi}{2}) = 0$. We substitute $x = \frac{\pi}{2}$ and $y = 0$ into the general solution to find the value of the constant $C$. Since $\csc(\frac{\pi}{2}) = 1$, we get $0 = \frac{\pi^2}{4} + C$, which means $C = -\frac{\pi^2}{4}$.

Step 9: Substitute C back into the general solution

Finally, we substitute the value of $C$ back into the general solution for $y$. This gives us the particular solution to the differential equation that satisfies the given initial condition.

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