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If $m$ and $n$ are respectively the order and the degree of the differential equation representing the family of curves $y^2-5 a x-5 a^{\frac{3}{2}}=0(a>0$ is a parameter $)$, then the value of $m-n$ is
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Verified Answer
The correct answer is:
-2
Given family of curves is
now differentiating w.r.t. $x$
$2 y y^{\prime}-5 a=0$
Putting value of a in equation (i)
$$
\begin{aligned}
& y^2-5 \cdot\left(\frac{2}{5} y y^{\prime}\right) x-5\left(\frac{2}{5} y y^{\prime}\right)^{\frac{3}{2}}=0 \\
& \Rightarrow y^2-2 y y^{\prime} x=5\left(\frac{2}{5} y y^{\prime}\right)^{\frac{3}{2}}
\end{aligned}
$$
squarring both side
$$
\begin{aligned}
& \left(\mathrm{y}^2-2 \mathrm{yy}^{\prime} \mathrm{x}\right)^2=5^2\left(\frac{2}{5} \mathrm{yy}^{\prime}\right)^3 \\
& \Rightarrow \mathrm{m}=1, \mathrm{n}=3 \\
& \mathrm{~m}-\mathrm{n}=-2
\end{aligned}
$$
now differentiating w.r.t. $x$
$2 y y^{\prime}-5 a=0$
Putting value of a in equation (i)
$$
\begin{aligned}
& y^2-5 \cdot\left(\frac{2}{5} y y^{\prime}\right) x-5\left(\frac{2}{5} y y^{\prime}\right)^{\frac{3}{2}}=0 \\
& \Rightarrow y^2-2 y y^{\prime} x=5\left(\frac{2}{5} y y^{\prime}\right)^{\frac{3}{2}}
\end{aligned}
$$
squarring both side
$$
\begin{aligned}
& \left(\mathrm{y}^2-2 \mathrm{yy}^{\prime} \mathrm{x}\right)^2=5^2\left(\frac{2}{5} \mathrm{yy}^{\prime}\right)^3 \\
& \Rightarrow \mathrm{m}=1, \mathrm{n}=3 \\
& \mathrm{~m}-\mathrm{n}=-2
\end{aligned}
$$
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