Determine the differential equation of parabolas with foci at origin and axes along X-axis.
Hint:y2=4a(x−a)
Consider the equitation of parabolas with foci at origin and axes along x−axis.
y2=4a(x−a) …….(1)
Differentiate with respect to x,we get
2ydydx=4a(1−0)
a=12ydydx
Put this value in equation 1st, we get
y2=4×12ydydx⎛⎜ ⎜ ⎜⎝x−12ydydx⎞⎟ ⎟ ⎟⎠
ydydxy2=2⎛⎜ ⎜ ⎜⎝x−12ydydx⎞⎟ ⎟ ⎟⎠
y3dydx=2⎛⎜ ⎜ ⎜⎝2xydydx−12ydydx⎞⎟ ⎟ ⎟⎠
y4(dydx)2=2xydydx−1
Hence ,this is the answer .