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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
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Solution
(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.
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