Question

Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.

Answer 1

The equation of the parabola having vertex at origin and axis along the positive direction of x-axis is given by
y² =4ax .....(1)
Since there is only one parameter, so we differentiate it only once.
Differentiating with respect to x, we get
$2y\frac{dy}{dx}=4a$
Substituting the value of 4a in (1), we get
$$\begin{gathered} y^2=2y\frac{dy}{dx}\times x \\ \Rightarrow y^2=2xy\frac{dy}{dx} \\ \Rightarrow y^2-2xy\frac{dy}{dx}=0 \end{gathered}$$