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siriusmart

[2025/01/15] Twice derivative of function (lemmy.world)

submitted 4 weeks ago by siriusmart to c/dailymaths

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Question

Find the function f such that f'' = -f
Show that the function can be written in format f = A sin(x + B)

Extension

Find a function g such that g'' = C * -g and

g(0) = D
g'(0) = 0 where C and D are arbitrary constants.

For those who have finished the extension

Congratulations, you have just derived the formula for objects in simple harmonic motions (physics).

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[–] siriusmart 8 points 4 weeks ago* (last edited 4 weeks ago) (1 children)

when i first showed this question to my friend, he told me that f(x) = 0 satisfies the condition. It is incredible that he said this without delay (i have not thought of that possibility at all), but at the same time it made me wanted to explode,

[–] [email protected] 2 points 4 weeks ago* (last edited 4 weeks ago)

That's why one usually searches for non trivial solution of differential equations. The trivial solution f(x)=0, in physics: "nothing happens", is usually not of interest, as we know by experience that something must happen to a system when some initial action is performed, yet mathematically it is always there.