Y'(x)=sin(x)/cos(y) differential equation

To solve the differential equation \( Y'(x) = \frac{\sin(x)}{\cos(Y(x))} \), let’s rewrite it in a separable form:

1. Start by rewriting \( Y'(x) = \frac{dY}{dx} \) as:
   \[
   \frac{dY}{dx} = \frac{\sin(x)}{\cos(Y)}
   \]

2. Separate variables, moving all \(Y\)-related terms to one side and \(x\)-related terms to the other:
   \[
   \cos(Y) \, dY = \sin(x) \, dx
   \]

3. Now, integrate both sides:
   \[
   \int \cos(Y) \, dY = \int \sin(x) \, dx
   \]

4. Integrating each side:
   - The integral of \(\cos(Y)\) with respect to \(Y\) is \(\sin(Y)\).
   - The integral of \(\sin(x)\) with respect to \(x\) is \(-\cos(x)\).
   
   Thus, we get:
   \[
   \sin(Y) = -\cos(x) + C
   \]
   where \(C\) is the constant of integration.

5. Solving for \(Y\), we obtain the implicit solution:
   \[
   Y = \arcsin(-\cos(x) + C)
   \]

This is the general solution of the differential equation. 

Would you like more details on this solution or have any questions?

Here are five related questions to extend your understanding:

1. How do we determine \(C\) if an initial condition is given?
2. What is the domain of the solution based on the properties of \(\arcsin\)?
3. How can this differential equation be solved using numerical methods?
4. How would the solution change if \(Y'(x)\) had a different trigonometric expression?
5. What are some applications of such separable differential equations in real-world contexts?

**Tip:** When solving separable differential equations, always check if both sides are easily integrable before separating.

Y'(x) = sin(x) / cos(y) differential equation

Explore how to solve the differential equation Y'(x) = sin(x) / cos(Y) using separation of variables. This solution involves trigonometric integration and yields an implicit solution in terms of the arcsin function, suitable for advanced high school students in Grades 11-12.

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