Answer 1

The equation of the parbola whose axis is the X-axis is y² = 4a(x - h),     ....(1)

where a and h are arbitrary constants.

Differentiating (1) w.r.t. x, we get

`"2y"("dy"/"dx") = 4"a"(1 - 0)`

∴ y`"dy"/"dx" = "2a"`

Differentiating again w.r.t. x, we get

`"y" * "d"/"dx"("dy"/"dx") + "dy"/"dx" * "dy"/"dx" = 0`

∴ `"y"("d"^2"y")/"dx"^2 + ("dy"/"dx")^2 = 0`

This is the required D.E.