Sol. The equation of the given family of curves is

y = Cx 2 ... (i)

Differentiating (i) with respect to x, we get

= 2Cx ... (ii)

Eliminating C between (i) and (ii), we obtain

y = x 2 ⇒ 2y = x ... (iii)

This is the differential equation of the family of curves given in (i).

The differential equation of the orthogonal trajectories of (i) is obtained by replacing by – in (iii).

Replacing by in (iii), we obtain

2y = – x

⇒ 2y dy = –x dx

On integrating, we obtain

y

2 = – + C

⇒ x 2 + 2y 2 = 2C

This is the required family of orthogonal trajectories.