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

\[y^2 = 2y\frac{dy}{dx} \times x\]
\[ \Rightarrow y^2 = 2xy\frac{dy}{dx}\]
\[ \Rightarrow y^2 - 2xy\frac{dy}{dx} = 0\]
\[\]