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Question
Form the differential equation of all parabolas whose axis is the X-axis.
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Solution
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.
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