Question

Consider the differential equation \[ \sin\left(\frac{y}{x}\right)\frac{dy}{dx} + 1 = \frac{y}{x}\sin\left(\frac{y}{x}\right) \] with \( y(1) = \frac{\pi}{2} \). Let \[ \alpha = \cos\left(\frac{y(e^{12})}{e^{12}}\right). \] If \( r \) is the radius of the circle \[ x^2 + y^2 - 2px + 2py + \alpha + 2 = 0, \] then the number of integral values of \( p \) is:

• A. 11
• B. 12
• C. 13
• D. 15

Hint:

In homogeneous equations where the function is \( y/x \), the term \( v \sin v \) often cancels out after substitution, leaving a simple separable equation.

Answer 1

The Correct Option is C