The solution of the differential equation \(\displaystyle\frac{dy}{dx}=\frac{ax + c}{by + d}\) represents a circle when
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(a = -b\)
The equation is in variables separable form:
\((by + d)\,dy = (ax + c)\,dx\)
Integrating both sides:
\(\displaystyle \frac{by^2}{2} + dy = \frac{ax^2}{2} + cx + K\)
\(\displaystyle \frac{a}{2}x^2 - \frac{b}{2}y^2 + cx - dy + K = 0\)
A second-degree equation with no xy term is a circle when the coefficients of \(x^2\) and \(y^2\) are equal and non-zero:
\(\displaystyle \frac{a}{2} = -\frac{b}{2}\)
\(a = -b\)