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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a sin (x + b)
Solution
y = a sin (x + b)
∴ `"dy"/"dx" = "a" "d"/"dx" [sin ("x + b")]`
`= "a" cos ("x + b") - "d"/"dx" ("x + b")`
= a cos (x + b) × (1 + 0)
= a cos (x + b)
and `("d"^2 "y")/"dx"^2 = "a" "d"/"dx"[cos ("x + b")]`
`= "a" [- sin ("x + b")] * "d"/"dx"("x + b")`
`= - "a" sin ("x + b") xx (1 + 0)`
∴ `("d"^2 "y")/"dx"^2 = - "y"` .....[By (1)]
∴ `("d"^2 "y")/"dx"^2 + "y" = 0`
This is the required D.E.
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