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(D) Given family of curves is $y^2 - 5ax - 5a^{3/2} = 0$  ... $(i)$Differentiating with respect to $x$:$2yy' - 5a = 0 \Rightarrow a = \frac{2}{5}yy'$S

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-$2$

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(D) Given family of curves is $y^2 - 5ax - 5a^{3/2} = 0$  ... $(i)$Differentiating with respect to $x$:$2yy' - 5a = 0 \Rightarrow a = \frac{2}{5}yy'$Substituting the value of $a$ in equation $(i)$:$y^2 - 5\left(\frac{2}{5}yy'\right)x - 5\left(\frac{2}{5}yy'\right)^{3/2} = 0$$y^2 - 2yy'x = 5\left(\frac{2}{5}yy'\right)^{3/2}$Squaring both sides:$(y^2 - 2yy'x)^2 = 25 \cdot \left(\frac{2}{5}yy'\right)^3$$(y^2 - 2yy'x)^2 = 25 \cdot \frac{8}{125} (yy')^3$$(y^2 - 2yy'x)^2 = \frac{8}{5} (yy')^3$The highest order derivative is $y'$, so the order $m = 1$.The power of the highest order derivative is $3$, so the degree $n = 3$.Therefore, $m - n = 1 - 3 = -2$.

If $m$ and $n$ are respectively the order and the degree of the differential equation representing the family of curves $y^2 - 5ax - 5a^{3/2} = 0$ ($a > 0$ is a parameter), then the value of $m - n$ is

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