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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = A cos (log x) + B sin (log x)
Solution
y = A cos (log x) + B sin (log x) ...(1)
Differentiating w.r.t. x, we get
`"dy"/"dx" = - "A sin" ("log x")*"d"/"dx" ("log x") + "B cos" ("log x")*"d"/"dx" ("log x")`
`= (- "A sin" ("log x"))/"x" + ("B cos" (log "x"))/"x"`
∴ `"x" "dy"/"dx"` = – A sin (log x) + B cos (log x)
Differentiating again w.r.t. x, we get
`"x" ("d"^2"y")/"dx"^2 + "dy"/"dx" = (- "A cos" ("log x"))/"x" + ("B sin" (log "x"))/"x"`
∴ `"x"^2 ("d"^2"y")/"dx"^2 + "x""dy"/"dx"` = – [A cos (log x) + B sin (log x)] = – y .....[By (1)]
∴ `"x"^2 ("d"^2"y")/"dx"^2 + "x""dy"/"dx" + "y"` = 0 is the required D.E.
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