Question

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

`"y"^2 = "a"("b - x")("b + x")`

Answer 1

`"y"^2 = "a"("b - x")("b + x") = "a"("b"^2 - "x"^2)`

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

`"2y" "dy"/"dx" = "a" (0 - 2"x") = - 2 "ax"`

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

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

`"y" * "d"/"dx" ("dy"/"dx")^2+ "dy"/"dx" * "dy"/"dx" = - "a" xx 1`

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

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

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

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

This is the required D.E.