The differential equation of the family of parabolas with focus at the origin and the $\mathrm{X}$ - axis as…
The differential equation of the family of parabolas with focus at the origin and the $\mathrm{X}$ - axis as axis, is
- $-y\left(\frac{d y}{d x}\right)^2=2 x \frac{d y}{d x}-y$
- $y\left(\frac{d y}{d x}\right)^2+2 x y \frac{d y}{d x}+y=0$
- $y\left(\frac{d y}{d x}\right)^2+4 x \frac{d y}{d x}=4 x y$
- $y\left(\frac{d y}{d x}\right)^2+y=2 x y \frac{d y}{d x}$
Solution
Equation of parabola is $y^2=4 a(x+a)$
$2 y \frac{d y}{d x}=4 a \Rightarrow a=\frac{y \frac{d y}{d x}}{2}$
Substituting value of ' $a$ ', we get
$\begin{aligned}
& y^2=4\left(\frac{y \frac{d y}{d x}}{2}\right)\left[x+\left(\frac{y \frac{d y}{d x}}{2}\right)\right]=\left(2 y \frac{d y}{d x}\right)\left(\frac{2 x+y \frac{d y}{d x}}{2}\right) \\
& 2 y^2=\left(2 y \frac{d y}{d x}\right)\left(2 x+y \frac{d y}{d x}\right)=4 x y \frac{d y}{d x}+2 y^2\left(\frac{d y}{d x}\right)^2 \\
& \therefore \quad y=2 x \frac{d y}{d x}+y\left(\frac{d y}{d x}\right)^2
\end{aligned}$
Asked in: MHT CET 2021 (23 Sep Shift 1)
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