Question

Obtain the differential equation representing the family of parabola having vertex at origin and axis along the positive direction of $x$-axis.

Answer 1

Equation of parabola having a vertex at origin and axis along the positive direction of $x$-axis is

$y^2=4ax$

Differentiating both sides w.r.t $x$ we get

$2y\frac{dy}{dx}=4a$

$\Rightarrow a=\frac{y}{2}\frac{dy}{dx}$

Substituting for $a$ in $y^2=4ax$ we get

$y^2=4\left(\frac{y}{2}\frac{dy}{dx}\right)x$

$\Rightarrow y^2=2xy\frac{dy}{dx}$

$\Rightarrow y^2-2xy\frac{dy}{dx}=0$ is the required differential equation