Question

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

(y - a)² = b(x + 4)

Answer 1

(y - a)² = b(x + 4)     ....(1)

Differentiating both sides w.r.t. x, we get

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

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

∴ `2("y - a") "dy"/"dx" = "b"`

∴ `2("y - a") "dy"/"dx" = ("y - a")^2/("x + 4")`   ....[By (1)]

`2 ("x + 4") "dy"/"dx" = "y - a"`

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

`2 [("x + 4") * "d"/"dx"("dy"/"dx") + "dy"/"dx" * "d"/"dx" ("x + 4")] = "dy"/"dx" - 0`

∴ `2 [("x + 4") ("d"^2"y")/"dx"^2 + "dy"/"dx" xx (1 + 0)] = "dy"/"dx"`

∴ `2("x + 4") ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" - "dy"/"dx" = 0`

∴ `2("x + 4") ("d"^2"y")/"dx"^2 + "dy"/"dx" = 0`

This is the required D.E.