The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{a + bx}{c + dy}$ represents a family of circles centered at the origin if...

The equation of a family of circles centered at the origin is given by $x^2 + y^2 = r^2$, where $r$ is an arbitrary constant.

Differentiating this equation with respect to $x$, we get:

$2x + 2y \dfrac{dy}{dx} = 0$

$\dfrac{dy}{dx} = -\dfrac{x}{y}$

The given differential equation is:

$\dfrac{dy}{dx} = \dfrac{a + bx}{c + dy}$

For this to represent the family of circles centered at the origin, it must be identical to $\dfrac{dy}{dx} = -\dfrac{x}{y}$.

Comparing the two equations, we get:

$\dfrac{a + bx}{c + dy} = \dfrac{-x}{y}$

This requires $a = 0$ and $c = 0$. The equation then becomes:

$\dfrac{bx}{dy} = -\dfrac{x}{y}$

$\dfrac{b}{d} = -1$

$b = -d \Rightarrow b + d = 0$

Thus, the required conditions are $a = c = 0$ and $b + d = 0$.

Answer: $a = c = 0,\ b + d = 0$