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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Solution
(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.
Notes
The answer in the textbook is incorrect.
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