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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
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Solution
Ax2 + By2 = 1
Differentiating both sides w.r.t. x, we get
`"A" xx "2x" + "B" xx "2y" "dy"/"dx" = 0`
∴ `"Ax" + "By" "dy"/"dx" = 0` ....(1)
Differentiating again w.r.t. x, we get
`"A" xx 1 + "B" ["y" "d"/"dx" ("dy"/"dx") + "dy"/"dx"*"dy"/"dx"] = 0`
∴ `"A + B" ["y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2] = 0`
∴ `"A" = - "B"["y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2]`
Substituting the value of A in (1), we get
`- "B x"["y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2] + "B y" "dy"/"dx" = 0`
∴ `- "x" ["y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2] + "y" "dy"/"dx" = 0`
∴ `- "xy" ("d"^2"y")/"dx"^2 - "x" ("dy"/"dx")^2 + "y" "dy"/"dx" = 0`
∴ `"xy" ("d"^2"y")/"dx"^2 + "x" ("dy"/"dx")^2 - "y" "dy"/"dx" = 0`
This is the required D.E.
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