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(C) The equation of the family of parabolas with focus at the origin and $X$-axis as its axis is given by $y^2 = 4a(x+a) = 4ax + 4a^2$. Differentiatin

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$3$

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(C) The equation of the family of parabolas with focus at the origin and $X$-axis as its axis is given by $y^2 = 4a(x+a) = 4ax + 4a^2$. Differentiating with respect to $x$,we get $2y \frac{dy}{dx} = 4a$,which implies $a = \frac{1}{2} y \frac{dy}{dx}$. Substituting the value of $a$ into the original equation: $y^2 = 4 \left( \frac{1}{2} y \frac{dy}{dx} ight) x + 4 \left( \frac{1}{2} y \frac{dy}{dx} ight)^2$ $y^2 = 2xy \frac{dy}{dx} + y^2 \left( \frac{dy}{dx} ight)^2$. The highest order derivative is $\frac{dy}{dx}$,so the order $m = 1$. The power of the highest order derivative is $2$,so the degree $n = 2$. Therefore,$mn - m + n = (1 	imes 2) - 1 + 2 = 2 - 1 + 2 = 3$.

If $m$ and $n$ are respectively the order and degree of the differential equation of the family of parabolas with focus at the origin and $X$-axis as its axis,then $m n-m+n=$

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