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(C) The equation of a family of circles with center $(0, b)$ on the $Y$-axis and radius $a$ is given by $x^2 + (y - b)^2 = a^2$.Differentiating with r

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$xy_2 = y_1(y_1^2 + 1)$

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(C) The equation of a family of circles with center $(0, b)$ on the $Y$-axis and radius $a$ is given by $x^2 + (y - b)^2 = a^2$.Differentiating with respect to $x$:$2x + 2(y - b)y_1 = 0$$x + (y - b)y_1 = 0$ ... $(i)$Differentiating again with respect to $x$:$1 + (y - b)y_2 + y_1^2 = 0$$(y - b)y_2 = -(1 + y_1^2)$$y - b = -\frac{1 + y_1^2}{y_2}$ ... $(ii)$Substituting $(ii)$ into $(i)$:$x + \left(-\frac{1 + y_1^2}{y_2}\right)y_1 = 0$$x - \frac{y_1(1 + y_1^2)}{y_2} = 0$$xy_2 = y_1(1 + y_1^2)$

The differential equation representing the family of circles having their centres on the $Y$-axis is (where $y_1 = \frac{dy}{dx}$ and $y_2 = \frac{d^2y}{dx^2}$):

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