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The order of the differential equation corresponding to the family of parabolas whose axes are along the $X$-axis and whose foci are at the origin, is
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The correct answer is:
1
The equation of family of parabolas with focus and the $x$-axis as axis is
Differentiating w.r.t. $x$, we get
$$
2 y \frac{d y}{d x}=4 a \Rightarrow a=\frac{y}{2} \frac{d y}{d x}
$$
Substituting the value of $a$ in Eq. (i), we get
$$
\begin{aligned}
& y^2=2 y \frac{d y}{d x}\left(x-\frac{y}{2}\right) \frac{d y}{d x} \\
\Rightarrow & y^2=y \frac{d y}{d x}\left(2 x-y \frac{d y}{d x}\right) \\
\Rightarrow & y\left(\frac{d y}{d x}\right)^2-2 x \frac{d y}{d x}+y=0
\end{aligned}
$$
It is clear from the differential equation is that the orders of equation is 1 .
Differentiating w.r.t. $x$, we get
$$
2 y \frac{d y}{d x}=4 a \Rightarrow a=\frac{y}{2} \frac{d y}{d x}
$$
Substituting the value of $a$ in Eq. (i), we get
$$
\begin{aligned}
& y^2=2 y \frac{d y}{d x}\left(x-\frac{y}{2}\right) \frac{d y}{d x} \\
\Rightarrow & y^2=y \frac{d y}{d x}\left(2 x-y \frac{d y}{d x}\right) \\
\Rightarrow & y\left(\frac{d y}{d x}\right)^2-2 x \frac{d y}{d x}+y=0
\end{aligned}
$$
It is clear from the differential equation is that the orders of equation is 1 .
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