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Question
The differential equation obtained from the function y = a(x − a)2 is ____________.
Options
`8"y"^2 = ("dy"/"dx")^2 ["x" + 1/(4"y") ("dy"/"dx")^2]^2`
`4"y"^2 = ("dy"/"dx")^2 ["x" - 1/(4"y") ("dy"/"dx")^2]^2`
`2"y"^2 = ("dy"/"dx")^2 ["x" - 1/(4"y") ("dy"/"dx")^2]^2`
`8"y"^2 = ("dy"/"dx")^2 ["x" - 1/(4"y") ("dy"/"dx")^2]^2`
Solution
The differential equation obtained from the function y = a(x − a)2 is `underline(4"y"^2 = ("dy"/"dx")^2 ["x" - 1/(4"y") ("dy"/"dx")^2]^2)`.
Explanation:
We have,
y = a(x − a)2 ...............(i)
`"dy"/"dx"` = 2a(x − a)
`("dy"/"dx")^2` = 4a2 (x − a)2 .........(ii)
From Eqs. (i) and (ii), we get
`(("dy"/"dx")^2)/"y" = (4"a"^2 ("x" - "a")^2)/("a" ("x" - "a")^2)` = 4a
∴ a = `(("dy"/"dx")^2)/(4"y")`
Putting, the value of a in Eq. (i), we get
y = `(("dy"/"dx")^2)/(4"y") ["x" - 1/(4"y") ("dy"/"dx")^2]^2`
4y2 = `("dy"/"dx")^2 ["x" - 1/(4"y") ("dy"/"dx")^2]^2`
