Question

If the solution of the differential equation \[ \frac{dy}{dx} = \frac{ax + 3}{2y + 5} \] represents a circle, then $a$ is equal to:

• A. 3
• B. -2
• C. -3
• D. 5

Answer 1

The Correct Option is C

For the given differential equation:

\[ \frac{dy}{dx} = \frac{ax + 3}{2y + 5}, \]

the solution represents a circle if the equation can be transformed into the standard form of a circle equation:

\[ (x - h)^2 + (y - k)^2 = r^2, \]

where \( h, k \) are the coordinates of the circle’s center and \( r \) is its radius.

To ensure the solution represents a circle, the differential equation must be separable and upon integration, should result in an equation of the form of a circle. This implies that the coefficient \( a \) must satisfy certain conditions.

After analyzing and comparing with known forms and conditions required for a circle, it can be determined that:

\[ a = -3. \]