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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y2 = (x + c)3
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Solution
y2 = (x + c)3 ...(1)
Differentiating w.r.t. x, we get
`"2y" "dy"/"dx" = 3("x + c")^2 * (1) = 3("x + c")^2`
∴ `("x + c")^2 = "2y"/3 * "dy"/"dx"`
∴ `("x + c")^6 = ("2y"/3 * "dy"/"dx")^3`
∴ `("y"^2)^2 = "8y"^3/27 * ("dy"/"dx")^3` ....[By (1)]
∴ `"27y"^4 = "8y"^3("dy"/"dx")^3`
∴ `"27y" = 8("dy"/"dx")^3`
∴ `8("dy"/"dx")^3 - 27"y" = 0`
This is the required D.E.
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