The solution of the differential equation $\frac{dy}{dx}=\frac{ax + c}{by + d}$ represents a circle when

$a =-b$

The correct answer is Option (1) → $a =-b$

Given differential equation:

$\frac{dy}{dx} = \frac{ax + c}{by + d}$

Separate variables:

$(by + d) dy = (ax + c) dx$

Integrate both sides:

$\int (by + d)\,dy = \int (ax + c)\,dx$

$\frac{b}{2}y^2 + dy = \frac{a}{2}x^2 + cx + C$

Rewriting:

$\frac{a}{2}x^2 + cx - \frac{b}{2}y^2 - dy + C = 0$

To represent a circle, the general second-degree equation must satisfy:

• Both $x^2$ and $y^2$ terms must be present with equal coefficients.
• No $xy$ term.

In our case, coefficients of $x^2$ and $y^2$ are:

$\frac{a}{2}$ and $-\frac{b}{2}$

For them to be equal (same coefficient and sign):

$\frac{a}{2} = -\frac{b}{2} \Rightarrow a = -b$