The differential equation of all circles passing through the origin and having their centres on the x-axis is :

Question

Accepted Answer

To determine the differential equation of all circles passing through the origin and having their centers on the x-axis, we start by considering the general equation of a circle with center at $(a, 0)$ and radius $r$:

$$
(x - a)^2 + y^2 = r^2
$$

Since the circle passes through the origin $(0, 0)$, substituting these coordinates into the equation gives:

$$
a^2 + 0^2 = r^2 \implies r^2 = a^2
$$

Thus, the equation of the circle becomes:

$$
(x - a)^2 + y^2 = a^2
$$

Expanding this equation, we have:

$$
x^2 - 2ax + a^2 + y^2 = a^2
$$

Simplifying, we get:

$$
x^2 + y^2 = 2ax
$$

To find the differential equation, we differentiate both sides with respect to $x$:

$$
\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(2ax)
$$

Using the chain rule, the left side becomes:

$$
2x + 2y\frac{dy}{dx}
$$

The right side differentiates to:

$$
2a
$$

Equating the derivatives, we have:

$$
2x + 2y\frac{dy}{dx} = 2a
$$

Since $2a = \frac{x^2 + y^2}{x}$ from the original circle equation, substitute back:

$$
2x + 2y\frac{dy}{dx} = \frac{x^2 + y^2}{x}
$$

Multiply through by $x$ to clear the fraction:

$$
2x^2 + 2xy\frac{dy}{dx} = x^2 + y^2
$$

Rearranging terms gives:

$$
y^2 = x^2 + 2xy\frac{dy}{dx}
$$

This matches Option 3, confirming it as the correct answer.

**Explanation of Incorrect Options:**

1. **Option 1: $x^{2}=y^{2}+xy\frac{dy}{dx}$**
   - This equation does not correctly represent the relationship derived from the circle's equation. The differential term $xy\frac{dy}{dx}$ is not sufficient to account for the $2xy\frac{dy}{dx}$ needed to balance the equation derived from the circle's properties.

2. **Option 2: $x^{2}=y^{2}+3xy\frac{dy}{dx}$**
   - This option incorrectly includes a factor of 3 in the differential term, which does not align with the derived equation. The factor should be 2, as shown in the correct derivation.

3. **Option 4: $y^{2}=x^{2}-2xy\frac{dy}{dx}$**
   - This option incorrectly places a negative sign in front of the differential term. The correct derivation shows that the term should be positive, as the relationship between the circle's geometry and its differential equation requires a positive contribution from the differential term.

Thus, Option 3 is the only choice that accurately represents the differential equation for circles passing through the origin with centers on the x-axis.