Obtain the general solutions of the given differential equations.  y¹= x sin² y

We are tasked with finding the general solution of the given differential equation:

\[
y' = x \sin^2 y
\]

This is a first-order, nonlinear ordinary differential equation (ODE). To solve it, we can use the method of separation of variables. The equation is separable because we can express all terms involving \(y\) on one side and all terms involving \(x\) on the other side.

### Step 1: Separate Variables
Rewriting the equation:
\[
\frac{dy}{dx} = x \sin^2 y
\]

We aim to separate \(dy\) and \(dx\):
\[
\frac{1}{\sin^2 y} dy = x dx
\]

### Step 2: Integrate Both Sides
Now, integrate both sides.

- For the left-hand side:
  \[
  \int \frac{1}{\sin^2 y} dy = \int \csc^2 y dy
  \]
  The integral of \(\csc^2 y\) is \(-\cot y\), so:
  \[
  -\cot y = \int \csc^2 y dy
  \]

- For the right-hand side:
  \[
  \int x dx = \frac{x^2}{2}
  \]

Thus, we have:
\[
-\cot y = \frac{x^2}{2} + C
\]
where \(C\) is the constant of integration.

### Step 3: Solve for \(y\)
Rearranging the equation to solve for \(y\):
\[
\cot y = -\left(\frac{x^2}{2} + C\right)
\]

Taking the arccotangent on both sides:
\[
y = \cot^{-1} \left(-\left(\frac{x^2}{2} + C\right)\right)
\]

Thus, the general solution is:
\[
y = \cot^{-1} \left(-\left(\frac{x^2}{2} + C\right)\right)
\]

### Final Solution:
The general solution to the differential equation \( y' = x \sin^2 y \) is:
\[
y = \cot^{-1} \left(-\left(\frac{x^2}{2} + C\right)\right)
\]

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Would you like further details on any part of the solution?

Here are 5 questions related to this topic:

1. What are the properties of the \(\cot^{-1}(x)\) function in calculus?
2. How do you verify if a solution satisfies the original differential equation?
3. What other methods exist for solving nonlinear ODEs?
4. What is the significance of the constant of integration in differential equations?
5. How would you apply initial conditions to find a particular solution?

**Tip**: Always check whether a differential equation is separable; this method simplifies many ODEs.

Obtain the general solutions of the given differential equation: y' = x sin² y

Explore the detailed step-by-step solution to the differential equation y' = x sin² y using separation of variables. This problem highlights key concepts in solving first-order nonlinear differential equations and trigonometric integrals. Suitable for undergraduate math students.

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