Question

The particular solution of the differential equation \[ \left( y + x \frac{dy}{dx} \right) \sin y = \cos x \quad \text{at} \quad x = 0 \, \text{is:} \]

• A. \( \sin x + \cos y = 1 \)
• B. \( \cos x - \sin y = 1 \)
• C. \( \sin x - \cos y = 1 \)
• D. \( \cos x + \sin y = 1 \)

Hint:

When solving first-order linear differential equations, ensure you substitute initial conditions early to simplify the solution process.

Answer 1

The Correct Option is A

Step 1: Solve the differential equation.
The given differential equation is: \[ \left( y + x \frac{dy}{dx} \right) \sin y = \cos x \] We want to find the particular solution, which involves solving for \( y \) in terms of \( x \). Step 2: Substitute \( x = 0 \) and \( y = 0 \).
At \( x = 0 \), substitute \( y = 0 \) and solve for the constants involved in the equation. \[ \sin(0) + \cos(0) = 1 \] Thus, the particular solution is \( \boxed{\sin x + \cos y = 1} \).