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dailymaths·Daily Maths Challengesbysiriusmart

[2025/01/15] Twice derivative of function

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.

::: spoiler For those who have finished the extension Congratulations, you have just derived the formula for objects in simple harmonic motions (physics). :::

View original on lemmy.world
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Comments4
siriusmart
lemmy.world

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,

8
Successful_Try543reply
feddit.org

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.

2
Successful_Try543
feddit.org

For simplicity, I restricted my solution to the 'extension' part and embedded the first part into it.
I hope it is OK to upload it here.
The task is a neat example to train the procedure of finding a solution to ODEs with initial value problems.

Thank you!

::: spoiler Solution.

:::

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[2025/01/15] Twice derivative of function | Spyke