UPSC NDA Math Model Paper 10

The equation of parabola having their vertices at the origin and foci on the x-axis is
y2=4ax ....(1)
Differentiating on both side, we get
2y

dy

dx

=4a
⇒y

dy

dx

=2a
⇒a=

y

2

dy

dx

Put this value in equation (1), we get
y2=4

y

2

dy

dx

x
⇒y2−2xy

dy

dx

=0
⇒y(y−2xy′)=0, Here y′=

dy

dx

∴y−2xy′=0
Hence, the differential equation to the family of parabolas having their vertices at the origin and foci on the x-axis is y - 2xy' = 0