Question

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

(y - a)² = 4(x - b)

Answer 1

(y - a)² = 4(x - b)

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

`2 ("y - a")*"d"/"dx"("y - a") = 4 "d"/"dx" ("x - b")`

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

∴ `2 ("y - a")"dy"/"dx" = 4`

∴ `("y - a")"dy"/"dx" = 2`     ....(1)

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

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

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

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

∴ `2/("dy"/"dx") * ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2 = 0`      .....[By (1)]

∴ `2 ("d"^2"y")/"dx"^2 + ("dy"/"dx")^3 = 0`

This is the required D.E.