# Obtain the differential equation by eliminating the arbitrary constants from the following equation: (y - a)2 = 4(x - b) - Mathematics and Statistics

Sum

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

(y - a)2 = 4(x - b)

#### 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.

Concept: Formation of Differential Equations
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