Question

The particular solution of the differential equation \[ \sin^2 y \frac{dx}{dy} + x = \cot y \quad \text{when} \quad x = 0 \quad \text{and} \quad y = \frac{3\pi}{4} \text{ is} \]

• A. \( x = 1 + \cot y \)
• B. \( xy = \cot (x + y) \)
• C. \( xy = \cot (x - y) \)
• D. \( y = 1 + \cot x \)

Hint:

When solving differential equations, always separate the variables and integrate, applying the given initial conditions to find the particular solution.

Answer 1

The Correct Option is A

Step 1: Separate the variables.
The given differential equation is: \[ \sin^2 y \frac{dx}{dy} + x = \cot y. \] Rearrange the equation to separate the variables: \[ \frac{dx}{dy} = \frac{\cot y - x}{\sin^2 y}. \]
Step 2: Solve the equation.
Now, integrate both sides of the equation. First, we solve for \( x \) by applying the given initial conditions \( x = 0 \) when \( y = \frac{3\pi}{4} \). After solving the equation, we get: \[ x = 1 + \cot y. \]
Step 3: Conclusion.
Thus, the particular solution is \( x = 1 + \cot y \), which corresponds to option (A).