Practicing Success
The differential equation of the family of parabola with focus at the origin and the x-axis as axis, is |
$y\left(\frac{d y}{d x}\right)^2+4 x \frac{d y}{d x}=4 y$ $y\left(\frac{d y}{d x}\right)^2=2 x \frac{d y}{d x}-y$ $y\left(\frac{d y}{d x}\right)^2+y=2 x y \frac{d y}{d x}$ $y\left(\frac{d y}{d x}\right)^2+2 x y \frac{d y}{d x}+y=0$ |
$y\left(\frac{d y}{d x}\right)^2=2 x \frac{d y}{d x}-y$ |
The equation of the family of parabolas with focus at the origin and the x-axis as axis is $y^2=4 a(x-a)$, where $a$ is a parameter. ......(i) Differentiating with respect to 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 (i), we get $y^2 =2 y \frac{d y}{d x}\left(x-\frac{y}{2} \frac{d y}{d x}\right)$ $\Rightarrow y^2 =y \frac{d y}{d x}\left(2 x-y \frac{d y}{d x}\right) \Rightarrow y\left(\frac{d y}{d x}\right)^2-2 x \frac{d y}{d x}+y=0$ This is the required differential equation. |