Question

The general solution of the differential equation \[ \frac{dy}{dx} + \sin(x + y) = \sin(x - y) \] is

• A. \( \log \tan \frac{y}{2} + \sin x = C \)
• B. \( \log \tan \frac{y}{2} + \log \sin x = C \)
• C. \( \tan \frac{y}{2} + \log \sin x = C \)
• D. None of these

Hint:

When solving differential equations, use separation of variables and integration to find the general solution.

Answer 1

The Correct Option is B

Step 1: Rearrange the equation.
Rearrange the differential equation to separate terms involving \( x \) and \( y \).
Step 2: Solve the differential equation.
Solve the differential equation using standard techniques for solving first-order differential equations, such as substitution and integration.
Step 3: Conclusion.
The general solution is \( \log \tan \frac{y}{2} + \log \sin x = C \). Final Answer: \[ \boxed{\log \tan \frac{y}{2} + \log \sin x = C} \]